1903:
1412:
2080:
1898:{\displaystyle {\begin{aligned}&{\frac {a}{b}}\cdot \left({\frac {c}{d}}+{\frac {e}{f}}\right)\\={}&{\frac {a}{b}}\cdot \left({\frac {c}{d}}\cdot {\frac {f}{f}}+{\frac {e}{f}}\cdot {\frac {d}{d}}\right)\\={}&{\frac {a}{b}}\cdot \left({\frac {cf}{df}}+{\frac {ed}{fd}}\right)={\frac {a}{b}}\cdot {\frac {cf+ed}{df}}\\={}&{\frac {a(cf+ed)}{bdf}}={\frac {acf}{bdf}}+{\frac {aed}{bdf}}={\frac {ac}{bd}}+{\frac {ae}{bf}}\\={}&{\frac {a}{b}}\cdot {\frac {c}{d}}+{\frac {a}{b}}\cdot {\frac {e}{f}}.\end{aligned}}}
44:
7456:
10171:
1914:
4055:
10601:
10014:
8364:
4777:
in modern terms. Kronecker's notion did not cover the field of all algebraic numbers (which is a field in
Dedekind's sense), but on the other hand was more abstract than Dedekind's in that it made no specific assumption on the nature of the elements of a field. Kronecker interpreted a field such as
2071:
coordinates), and the multiplication is – less intuitively – combining rotating and scaling of the arrows (adding the angles and multiplying the lengths). The fields of real and complex numbers are used throughout mathematics, physics, engineering, statistics, and many other scientific disciplines.
9701:
11024:
Dropping one or several axioms in the definition of a field leads to other algebraic structures. As was mentioned above, commutative rings satisfy all field axioms except for the existence of multiplicative inverses. Dropping instead commutativity of multiplication leads to the concept of a
2070:
given by the real numbers of their describing expression, or as the arrows from the origin to these points, specified by their length and an angle enclosed with some distinct direction. Addition then corresponds to combining the arrows to the intuitive parallelogram (adding the
Cartesian
8209:
5803:
9274:
7521:
form an ordered field that is not
Archimedean. It is an extension of the reals obtained by including infinite and infinitesimal numbers. These are larger, respectively smaller than any real number. The hyperreals form the foundational basis of
4860:
synthesized the knowledge of abstract field theory accumulated so far. He axiomatically studied the properties of fields and defined many important field-theoretic concepts. The majority of the theorems mentioned in the sections
11226:
5363:
9564:
4504:
does not split into linear factors in this smaller field. Elaborating further on basic field-theoretic notions, it can be shown that two finite fields with the same order are isomorphic. It is thus customary to speak of
1417:
4754:
By a field we will mean every infinite system of real or complex numbers so closed in itself and perfect that addition, subtraction, multiplication, and division of any two of these numbers again yields a number of the
547:, which are based on fields with additional structure. Basic theorems in analysis hinge on the structural properties of the field of real numbers. Most importantly for algebraic purposes, any field may be used as the
7491:
is required to have a least upper bound. Any complete field is necessarily
Archimedean, since in any non-Archimedean field there is neither a greatest infinitesimal nor a least positive rational, whence the sequence
1401:
1196:
Fields can also be defined in different, but equivalent ways. One can alternatively define a field by four binary operations (addition, subtraction, multiplication, and division) and their required properties.
5213:
9810:
10580:
can help to shape mathematical expectations, often first by understanding questions about function fields, and later treating the number field case. The latter is often more difficult. For example, the
6564:
7930:
7357:
8056:) correspond to each other. The first manifestation of this is at an elementary level: the elements of both fields can be expressed as power series in the uniformizer, with coefficients in
3172:
7929:, but is not complete. The completion of this algebraic closure, however, is algebraically closed. Because of its rough analogy to the complex numbers, it is sometimes called the field of
10921:
8359:{\displaystyle \operatorname {Gal} \left(\mathbf {Q} _{p}\left(p^{1/p^{\infty }}\right)\right)\cong \operatorname {Gal} \left(\mathbf {F} _{p}((t))\left(t^{1/p^{\infty }}\right)\right).}
2697:
In addition to familiar number systems such as the rationals, there are other, less immediate examples of fields. The following example is a field consisting of four elements called
10313:
3257:
3204:
8672:. Consequently, as can be shown, the zeros of the following polynomials are not expressible by sums, products, and radicals. For the latter polynomial, this fact is known as the
2062:
It is immediate that this is again an expression of the above type, and so the complex numbers form a field. Complex numbers can be geometrically represented as points in the
1205:, fields can be defined by two binary operations (addition and multiplication), two unary operations (yielding the additive and multiplicative inverses respectively), and two
11280:
to be local fields. On the other hand, these two fields, also called
Archimedean local fields, share little similarity with the local fields considered here, to a point that
5716:
2227:
9213:
9966:
8651:
2271:
8607:. By means of this correspondence, group-theoretic properties translate into facts about fields. For example, if the Galois group of a Galois extension as above is not
1225:. One may equivalently define a field by the same two binary operations, one unary operation (the multiplicative inverse), and two (not necessarily distinct) constants
2124:. For example, it was unknown to the Greeks that it is, in general, impossible to trisect a given angle in this way. These problems can be settled using the field of
11318:
are algebraically closed by Łoś's theorem. For the same reason, they both have characteristic zero. Finally, they are both uncountable, so that they are isomorphic.
4906:
is a set that is equipped with an addition and multiplication operation and satisfes all the axioms of a field, except for the existence of multiplicative inverses
10502:
can be reconstructed, up to isomorphism, from its field of functions. In higher dimension the function field remembers less, but still decisive information about
11798:
2136:. These numbers, endowed with the field operations of real numbers, restricted to the constructible numbers, form a field, which properly includes the field
2128:. Real constructible numbers are, by definition, lengths of line segments that can be constructed from the points 0 and 1 in finitely many steps using only
7210:
Since fields are ubiquitous in mathematics and beyond, several refinements of the concept have been adapted to the needs of particular mathematical areas.
4723:, in 1832, devised necessary and sufficient criteria for a polynomial equation to be algebraically solvable, thus establishing in effect what is known as
1406:
The abstractly required field axioms reduce to standard properties of rational numbers. For example, the law of distributivity can be proven as follows:
10753:. Studying arithmetic questions in global fields may sometimes be done by looking at the corresponding questions locally. This technique is called the
13485:
9696:{\displaystyle K_{n}^{M}(F)=F^{\times }\otimes \cdots \otimes F^{\times }/\left\langle x\otimes (1-x)\mid x\in F\smallsetminus \{0,1\}\right\rangle .}
13534:
Colloque d'algèbre supérieure, tenu à Bruxelles du 19 au 22 décembre 1956, Centre Belge de
Recherches Mathématiques Établissements Ceuterick, Louvain
11168:
5271:
8206:. Strikingly, the Galois groups of these two fields are isomorphic, which is the first glimpse of a remarkable parallel between these two fields:
1994:. Addition and multiplication of real numbers are defined in such a way that expressions of this type satisfy all field axioms and thus hold for
12837:"Die Struktur der absoluten Galoisgruppe 𝔭-adischer Zahlkörper. [The structure of the absolute Galois group of 𝔭-adic number fields]"
8151:
394:
13158:
12293:
3642:(and also with the multiplication), and is therefore a field homomorphism. The existence of this homomorphism makes fields in characteristic
1337:
1267:
Rational numbers have been widely used a long time before the elaboration of the concept of field. They are numbers that can be written as
2273:
is not a constructible number, which implies that it is impossible to construct with compass and straightedge the length of the side of a
5143:
4991:
such that all nonzero elements become invertible: forming the field of fractions, and forming residue fields. The field of fractions of
11063:
10198:
can be added and multiplied pointwise, e.g., the product of two functions is defined by the product of their values within the domain:
4025:) are fields with finitely many elements, whose number is also referred to as the order of the field. The above introductory example
1161:
An equivalent, and more succinct, definition is: a field has two commutative operations, called addition and multiplication; it is a
9279:
The Ax–Kochen theorem mentioned above also follows from this and an isomorphism of the ultraproducts (in both cases over all primes
10811:, i.e., ones with abelian Galois group, or equivalently the abelianized Galois groups of global fields. A classical statement, the
9720:
7708:
is as small as desired. The following table lists some examples of this construction. The fourth column shows an example of a zero
3705:, and is closed under addition, multiplication, additive inverse and multiplicative inverse of a nonzero element. This means that
6477:
10354:
2737:
in the axioms above). The field axioms can be verified by using some more field theory, or by direct computation. For example,
568:
10952:
In addition to the additional structure that fields may enjoy, fields admit various other related notions. Since in any field
13421:
13298:
13280:
13082:
13050:
13024:
12903:
12826:
12752:
12669:
12542:
12196:
12171:
12153:
55:
7282:
8574:
11008:
form a Field containing the reals, and would be a field except for the fact that they are a proper class, not a set. The
10572:). As for local fields, these two types of fields share several similar features, even though they are of characteristic
9995:
instead of fields) is much more complicated, because the above equation may have several or no solutions. In particular
5398:(as opposed to an arbitrary power series), the representation of fractions is less important in this situation, though.
7442:
whose value is greater than that element, that is, there are no infinite elements. Equivalently, the field contains no
4746:, which means "body" or "corpus" (to suggest an organically closed entity). The English term "field" was introduced by
4542:
Historically, three algebraic disciplines led to the concept of a field: the question of solving polynomial equations,
387:
4650:
13621:
13584:
13459:
13384:
13355:
13326:
13242:
13205:
13136:
13002:
12965:
12691:
12643:
12580:
12514:
12485:
12456:
12419:
12403:
12395:
12368:
10516:
attempts to identify the simplest (in a certain precise sense) algebraic varieties with a prescribed function field.
10443:
9844:
3133:
9707:
8488:
is an irreducible polynomial (as above). For such an extension, being normal and separable means that all zeros of
4948:(which means every element is invertible). Similarly, fields are the commutative rings with precisely two distinct
10882:
3259:
is an abelian group under multiplication (where 0 is the identity element of the addition), and multiplication is
13552:
8859:
7093:
has an algebraic closure, which is moreover unique up to (non-unique) isomorphism. It is commonly referred to as
17:
8098:, these fields are not isomorphic.) The following facts show that this superficial similarity goes much deeper:
12883:
12597:
11098:
10981:. In addition to division rings, there are various other weaker algebraic structures related to fields such as
8392:, together with the standard derivative of polynomials forms a differential field. These fields are central to
6971:
1217:). These operations are then subject to the conditions above. Avoiding existential quantifiers is important in
9366:
is fundamentally important: extending the case of finite Galois extensions outlined above, this group governs
5218:
It is straightforward to show that, if the ring is an integral domain, the set of the fractions form a field.
3263:
over addition. Some elementary statements about fields can therefore be obtained by applying general facts of
13709:
12771:
12623:
8378:
4704:
380:
7572:
with respect to the topology of the space. The topology of all the fields discussed below is induced from a
6984:
polynomial equation with complex coefficients has a complex solution. The rational and the real numbers are
6362:. Any finite extension is necessarily algebraic, as can be deduced from the above multiplicativity formula.
2120:
In antiquity, several geometric problems concerned the (in)feasibility of constructing certain numbers with
13704:
10266:
9999:
are much more difficult to solve than in the case of fields, even in the specially simple case of the ring
8397:
7446:(elements smaller than all rational numbers); or, yet equivalent, the field is isomorphic to a subfield of
3230:
3177:
35:
11125:" for denoting one part of a constant and for the additive inverses is justified by this latter condition.
8419:
4738:
introduced, for a set of real or complex numbers that is closed under the four arithmetic operations, the
543:
Fields serve as foundational notions in several mathematical domains. This includes different branches of
12766:
10812:
10758:
10122:
9996:
8422:
in the arithmetic operations of addition and multiplication. An important notion in this area is that of
6093:
4659:
4340:
471:
249:
9328:
In addition, model theory also studies the logical properties of various other types of fields, such as
5798:{\displaystyle \mathbf {R} /\left(X^{2}+1\right)\ {\stackrel {\cong }{\longrightarrow }}\ \mathbf {C} .}
4138:
The addition and multiplication on this set are done by performing the operation in question in the set
12386:
10761:
reduces the problem of finding rational solutions of quadratic equations to solving these equations in
10754:
10439:
8393:
7661:
in the sense that it is a real number that can be approximated arbitrarily closely by rational numbers
7269:
7057:
11967:
13714:
11328:
10876:
10493:
9823:
9269:{\displaystyle \operatorname {ulim} _{p\to \infty }{\overline {\mathbf {F} }}_{p}\cong \mathbf {C} .}
8578:
8447:
7618:
6811:
5901:
5628:
4886:
4625:) only yields two values. This way, Lagrange conceptually explained the classical solution method of
9875:. In higher degrees, K-theory diverges from Milnor K-theory and remains hard to compute in general.
8017:
These two types of local fields share some fundamental similarities. In this relation, the elements
2286:
13073:
11035:; sometimes associativity is weakened as well. The only division rings that are finite-dimensional
10986:
10702:
10045:
A widely applied cryptographic routine uses the fact that discrete exponentiation, i.e., computing
8807:
8673:
6862:
6675:
6027:
3413:
2202:
2121:
1218:
517:
475:
10508:. The study of function fields and their geometric meaning in higher dimensions is referred to as
10528:
8973:
7071:
4805:
4543:
972:
340:
9929:
8627:
4715:) cannot be solved algebraically; however, his arguments were flawed. These gaps were filled by
4450:, it can be checked case by case using the above multiplication table that all four elements of
2247:
1917:
The multiplication of complex numbers can be visualized geometrically by rotations and scalings.
13601:
13068:
10957:
10931:
10727:
10577:
10435:
10362:(a geometric object defined as the common zeros of polynomial equations) consists of ratios of
10182:
10178:
two (two handles). The genus can be read off the field of meromorphic functions on the surface.
9981:
9515:
9089:
if and only if it holds in any algebraically closed field of sufficiently high characteristic.
7577:
6718:. A similar construction can be carried out with a set of indeterminates, instead of just one.
5543:
5521:
4919:
4728:
4649:, Lagrange thus linked what eventually became the concept of fields and the concept of groups.
4428:
3297:, which consists of a single element; this guides any choice of the axioms that define fields.
2129:
2084:
1268:
1202:
1064:
626:
494:
479:
432:
13556:
12605:
4938:
In the hierarchy of algebraic structures fields can be characterized as the commutative rings
13638:
13480:
12704:
10860:
10782:
10586:
10513:
10337:
9495:
9356:
9189:
8624:
be expressed in terms of addition, multiplication, and radicals, i.e., expressions involving
8134:
7523:
7035:) and is algebraically closed (big enough to contain solutions of all polynomial equations).
4828:
4551:
3260:
2125:
2115:
2067:
857:
544:
12740:
12725:
11357:
Historically, division rings were sometimes referred to as fields, while fields were called
13631:
13431:
13365:
13252:
13215:
13181:
13146:
13092:
12975:
12941:
12913:
12868:
12848:
12761:
12653:
12555:
12524:
12466:
11246:
10341:
10125:, the multiplication in a finite field is replaced by the operation of adding points on an
9988:
9973:
9394:
8877:
7573:
6874:
4654:
3550:
3310:
3127:
2919:. In particular, one may deduce the additive inverse of every element as soon as one knows
2884:
548:
537:
529:
327:
319:
291:
286:
277:
234:
176:
59:
13674:
13594:
13514:
13469:
13439:
13394:
13336:
13034:
12807:
12718:
12708:
12590:
12429:
12352:
12242:
9475:
A description in terms of generators and relations is also known for the Galois groups of
8:
13012:
12610:(in German), vol. 1 (2nd ed.), Braunschweig, Germany: Friedrich Vieweg und Sohn
12378:
11135:
11076:
10532:
10509:
10246:
10175:
9062:
8892:
that are algebraically independent over the prime field. Two algebraically closed fields
8439:
8415:
7505:
is the unique complete ordered field, up to isomorphism. Several foundational results in
7273:
7020:
6878:
6807:
6359:
6216:
5382:
4949:
4630:
4043:
is the smallest field, because by definition a field has at least two distinct elements,
3264:
1162:
513:
444:
345:
335:
86:
78:
69:
12852:
13678:
13518:
13448:
13061:
12945:
12872:
12551:
12320:
12302:
12277:
12246:
11080:
10804:
10582:
10095:
9992:
9815:
9711:
8957:
8535:
8374:
8079:
6748:
6200:
6119:
Extensions whose degree is finite are referred to as finite extensions. The extensions
5813:
Fields can be constructed inside a given bigger container field. Suppose given a field
5038:
4945:
4646:
4626:
4547:
4099:
3788:
1185:
490:
151:
142:
100:
13067:, Lecture Notes in Logic, vol. 5 (2nd ed.), Association for Symbolic Logic,
12920:
Kiernan, B. Melvin (1971), "The development of Galois theory from
Lagrange to Artin",
12501:, London Mathematical Society Student Texts, vol. 3, Cambridge University Press,
6877:
if it does not have any strictly bigger algebraic extensions or, equivalently, if any
4720:
13682:
13662:
13617:
13580:
13522:
13502:
13455:
13417:
13401:
13380:
13372:
13351:
13343:
13322:
13312:
13307:
13294:
13276:
13263:
13238:
13201:
13132:
13108:
13078:
13046:
13020:
12998:
12961:
12949:
12899:
12876:
12822:
12795:
12748:
12687:
12665:
12639:
12576:
12538:
12510:
12481:
12452:
12415:
12399:
12391:
12364:
12250:
12230:
12192:
12167:
12149:
11250:
10808:
10186:
9819:
9507:
9503:
9333:
9329:
8989:
8862:, a far-reaching extension of Galois theory applicable to algebro-geometric objects.
8103:
7634:
7545:
7535:
7464:
7429:
7399:
7395:
7132:
7081:
7009:
6847:
has a transcendence basis. Thus, field extensions can be split into ones of the form
6703:
6655:
6581:
6178:
5233:
4843:
4774:
4766:
4739:
4716:
2727:
plays the role of the additive identity element (denoted 0 in the axioms above), and
2274:
1169:
as the additive identity; the nonzero elements are a group under multiplication with
968:
714:
416:
13172:
11079:
division algebra, but is not a division ring. This fact was proved using methods of
7264:. For example, the real numbers form an ordered field, with the usual ordering
5137:. The operation on the fractions work exactly as for rational numbers. For example,
4885:
redeveloped Galois theory from 1928 through 1942, eliminating the dependency on the
2079:
838:. These operations are required to satisfy the following properties, referred to as
13670:
13654:
13609:
13590:
13510:
13494:
13465:
13435:
13390:
13332:
13230:
13193:
13167:
13124:
13030:
12990:
12929:
12891:
12856:
12803:
12714:
12631:
12601:
12586:
12502:
12494:
12473:
12444:
12436:
12425:
12348:
12340:
12312:
12269:
12238:
12222:
10612:
10605:
10590:
10418:
10363:
10328:
10257:
8451:
8443:
8426:
8423:
8381:, i.e., allow to take derivatives of elements in the field. For example, the field
8203:
7789:
7120:
4903:
4797:
4735:
4708:
4578:
4184:
3554:
3221:
equipped with two operations denoted as an addition and a multiplication such that
2063:
1940:, with the usual operations of addition and multiplication, also form a field. The
1198:
1177:
1012:
723:
612:
580:
533:
509:
171:
12324:
12316:
5392:). Since any Laurent series is a fraction of a power series divided by a power of
4150:
and taking the remainder as result. This construction yields a field precisely if
3441:
satisfying this equation, the smallest such positive integer can be shown to be a
756:, that is, a correspondence that associates with each ordered pair of elements of
196:
13627:
13427:
13413:
13361:
13270:
13248:
13222:
13211:
13177:
13142:
13104:
13088:
12971:
12937:
12909:
12864:
12814:
12649:
12627:
12532:
12520:
12462:
11084:
10941:
10458:
10421:
dense subvariety. In other words, the function field is insensitive to replacing
10366:, i.e., ratios of polynomial functions on the variety. The function field of the
9555:
8886:
over its prime field. The latter is defined as the maximal number of elements in
8654:
7897:
7627:
is another field in which, informally speaking, the "gaps" in the original field
7136:
6050:
5490:
5243:
5029:
4801:
4687:
4363:
4098:
The simplest finite fields, with prime order, are most directly accessible using
3662:
2987:
1262:
604:
525:
459:
436:
263:
257:
244:
224:
215:
181:
118:
48:
13542:
10705:, which asserts the non-existence of rational nonzero solutions to the equation
9390:
7432:
is an ordered field such that for each element there exists a finite expression
840:
489:
are commonly used and studied in mathematics, particularly in number theory and
13608:, Graduate Texts in Mathematics, vol. 83 (2nd ed.), Springer-Verlag,
13476:
12683:
12615:
12561:
11221:{\displaystyle \left\langle F\smallsetminus \{0\},\cdot ,{}^{-1}\right\rangle }
11075:, for which multiplication is neither commutative nor associative, is a normed
11005:
10594:
10485:
10126:
9969:
8608:
7760:
7688:
7569:
7443:
7410:
7193:
7189:
6306:
5632:
5422:
5358:{\displaystyle \sum _{i=k}^{\infty }a_{i}x^{i}\ (k\in \mathbb {Z} ,a_{i}\in F)}
5263:
3096:
1984:
1941:
1926:
1123:
556:
467:
428:
305:
30:
This article is about an algebraic structure. For vector valued functions, see
13613:
13498:
13234:
13128:
12994:
12895:
12635:
12448:
7050:. The situation that the algebraic closure is a finite extension of the field
6618:(and similarly for higher exponents) do not have to be considered here, since
2839:(highlighted in red in the tables at the right) is also a field, known as the
1173:
as the multiplicative identity; and multiplication distributes over addition.
43:
13698:
13666:
13506:
13259:
12799:
12700:
12506:
12344:
12234:
12210:
12188:
12180:
11432:
11102:
11027:
10694:
10649:
10159:
10155:
8612:
8409:
7814:
7219:
5586:
5477:
5453:
5432:
5412:
4878:
4724:
4622:
3608:
3092:
2184:
926:
868:
564:
521:
502:
483:
452:
191:
156:
113:
13532:(1957), "Sur les analogues algébriques des groupes semi-simples complexes",
12960:, Fields Institute Monographs, vol. 12, American Mathematical Society,
10785:
studies the (unsolved) problem whether any finite group is the Galois group
10170:
9355:
For fields that are not algebraically closed (or not separably closed), the
8904:
are isomorphic precisely if these two data agree. This implies that any two
7548:, such that all operations of the field (addition, multiplication, the maps
7496:, every element of which is greater than every infinitesimal, has no limit.
13558:
Beiträge zur
Geometrie der Lage (Contributions to the Geometry of Position)
13529:
12288:
10997:
10781:
Unlike for local fields, the Galois groups of global fields are not known.
10698:
10536:
10524:
10373:
9977:
9511:
9192:
states that any first order statement that holds for all but finitely many
9132:
8988:
and conversely. The mathematical statements in question are required to be
8953:
8520:
7477:
7455:
7150:, is exceptionally simple. It is the union of the finite fields containing
4683:
4336:
4157:
4013:
3442:
3305:
3294:
2841:
2803:
2133:
560:
552:
498:
365:
296:
130:
31:
11331:
between smooth proper algebraic curves over an algebraically closed field
11062:(in which multiplication is non-commutative). This result is known as the
10956:, any field has at least two elements. Nonetheless, there is a concept of
10260:. In this case the ratios of two functions, i.e., expressions of the form
9968:
This immediate consequence of the definition of a field is fundamental in
9822:
with coefficients the given field. For example, the process of taking the
13409:
13153:
12989:, Graduate Texts in Mathematics, vol. 211 (3rd ed.), Springer,
12779:
11092:
11013:
10615:
are among the most intensely studied number fields. They are of the form
10431:
10340:, i.e., complex differentiable functions. Their ratios form the field of
9403:. This statement subsumes the fact that the only algebraic extensions of
9099:
8909:
8905:
8053:
7958:
7482:
7472:
7468:
6411:
are necessarily algebraic as well. Moreover, the degree of the extension
6208:
4967:
3966:
3947:
2143:
1932:
1922:
608:
563:, the siblings of the field of rational numbers, are studied in depth in
463:
440:
424:
408:
355:
350:
239:
229:
203:
13547:, Grundlehren der mathematischen Wissenschaften, vol. 328, Springer
13043:
Algebra, Volume II: Fields with
Structures, Algebras and Advanced Topics
13658:
13197:
12982:
12933:
12860:
12836:
12281:
12226:
12206:
11088:
11054:
10982:
10489:
9499:
7513:
6204:
5955:
5468:
4882:
4731:, but conceived neither an explicit notion of a field, nor of a group.
4367:
3943:
2165:
2104:
allows construction of the square root of a given constructible number
105:
13321:, Graduate Text in Mathematics, vol. 7 (2nd ed.), Springer,
6042:
can also be regarded from the opposite point of view, by referring to
1396:{\displaystyle {\frac {b}{a}}\cdot {\frac {a}{b}}={\frac {ba}{ab}}=1.}
12568:
12307:
10990:
10041:(red) is zero if there is a line (blue) passing through these points.
9436:, and that the Galois groups of these finite extensions are given by
8551:
4796:. Prior to this, examples of transcendental numbers were known since
3933:
1222:
1206:
360:
166:
123:
91:
13408:, Springer Monographs in Mathematics, Translated from the French by
12331:
Banaschewski, Bernhard (1992), "Algebraic closure without choice.",
12273:
10944:. For general number fields, no such explicit description is known.
10442:
and for the classification of algebraic varieties. For example, the
8567:
that preserve addition and multiplication and that send elements of
3987:(a prime number), the prime field is isomorphic to the finite field
12257:
12215:
11625:
11067:
9896:
8582:
8506:
has only simple zeros. The latter condition is always satisfied if
7709:
7506:
6006:
satisfying a certain property, for example the biggest subfield of
4220:
is not a field: the product of two non-zero elements is zero since
3301:
2175:
420:
161:
51:
7499:
Since every proper subfield of the reals also contains such gaps,
6000:. The compositum can be used to construct the biggest subfield of
5208:{\displaystyle {\frac {a}{b}}+{\frac {c}{d}}={\frac {ad+bc}{bd}}.}
5061:
is constructed from the integers. More precisely, the elements of
3969:
if it has no proper (i.e., strictly smaller) subfields. Any field
10438:
of varieties. It is therefore an important tool for the study of
10098:, which is the inverse operation, i.e., determining the solution
9203:, also holds for the ultraproduct. Applied to the above sentence
4881:, and thus the area of analysis, to purely algebraic properties.
4550:. A first step towards the notion of a field was made in 1770 by
3351:
2152:. Using the labeling in the illustration, construct the segments
1293:
448:
13272:
Proceedings of the International Congress of Mathematicians 2014
13112:
12784:"Über eine neue Begründung der Theorie der algebraischen Zahlen"
10008:
9506:. The cohomological study of such representations is done using
8826:
is a Galois extension if and only if there is an isomorphism of
7481:) and limits, which should exist, do exist. More formally, each
7001:
does not have any rational or real solution. A field containing
4727:
today. Both Abel and Galois worked with what is today called an
3080:
2993:
In addition, the following properties are true for any elements
2142:
of rational numbers. The illustration shows the construction of
1913:
13643:"Die allgemeinen Grundlagen der Galois'schen Gleichungstheorie"
13642:
12783:
12213:(1927), "Eine Kennzeichnung der reell abgeschlossenen Körper",
11009:
10256:
of functions, one must consider algebras of functions that are
9714:, relates this to Galois cohomology by means of an isomorphism
7131:
is usually rather implicit since its construction requires the
2824:
95:
27:
Algebraic structure with addition, multiplication, and division
5415:
into a field, a field can be obtained from a commutative ring
4054:
9805:{\displaystyle K_{n}^{M}(F)/p=H^{n}(F,\mu _{l}^{\otimes n}).}
9071:
is elementarily equivalent to any algebraically closed field
6012:, which is, in the language introduced below, algebraic over
4633:, which proceeds by reducing a cubic equation for an unknown
3998:
introduced below. Otherwise the prime field is isomorphic to
2244:
Not all real numbers are constructible. It can be shown that
9524:, can be reinterpreted as a Galois cohomology group, namely
9502:
are fundamental in many branches of arithmetic, such as the
8573:
to themselves). The importance of this group stems from the
5808:
11628:
Earliest Known Uses of Some of the Words of Mathematics (F)
10600:
10589:(open as of 2017) can be regarded as being parallel to the
10013:
9000:, the addition and multiplication). A typical example, for
6684:
since there is no polynomial equation with coefficients in
6559:{\displaystyle \sum _{k=0}^{n-1}a_{k}x^{k},\ \ a_{k}\in E.}
4962:. Fields are also precisely the commutative rings in which
4846:(1891) studied the field of formal power series, which led
4286:
being prime) constructed in this way is usually denoted by
2146:
of constructible numbers, not necessarily contained within
13293:, Lecture Notes in Mathematics, vol. 1999, Springer,
13192:, Lecture Notes in Mathematics, vol. 1093, Springer,
12607:
Vorlesungen über Zahlentheorie (Lectures on Number Theory)
9177:
since it behaves in several ways as a limit of the fields
7272:
states that a field can be ordered if and only if it is a
4823:
The first clear definition of an abstract field is due to
435:
are defined and behave as the corresponding operations on
13350:, Graduate Texts in Mathematics, vol. 67, Springer,
12620:
Commutative algebra with a view toward algebraic geometry
10405:, i.e., the field consisting of ratios of polynomials in
9883:
8912:
and the same characteristic are isomorphic. For example,
8202:), yields (infinite) extensions of these fields known as
7509:
follow directly from this characterization of the reals.
5536:
cannot be expressed as the product of two polynomials in
10960:, which is suggested to be a limit of the finite fields
8810:
is not usually a field. For example, a finite extension
8597:
and the set of intermediate extensions of the extension
3683:
that is a field with respect to the field operations of
3267:. For example, the additive and multiplicative inverses
8450:
shows that finite separable extensions are necessarily
4682:
and, again using modern language, the resulting cyclic
3126:
form an abelian group under multiplication, called the
13118:
13059:
Marker, David; Messmer, Margit; Pillay, Anand (2006),
12573:
Modern algebra and the rise of mathematical structures
11428:
9077:
of characteristic zero. Moreover, any fixed statement
7352:{\displaystyle x_{1}^{2}+x_{2}^{2}+\dots +x_{n}^{2}=0}
583:
defined on that set: an addition operation written as
13536:, Paris: Librairie Gauthier-Villars, pp. 261–289
13156:(1893), "A doubly-infinite system of simple groups",
11171:
10885:
10269:
9932:
9723:
9567:
9216:
8630:
8212:
7285:
6480:
5719:
5681:, which implies that the map that sends a polynomial
5274:
5146:
4979:, there are two ways to construct a field related to
3322:
In addition to the multiplication of two elements of
3233:
3180:
3136:
2250:
2205:
1415:
1340:
5406:
In addition to the field of fractions, which embeds
3470:
since (in the notation of the above addition table)
603:, both of which behave similarly as they behave for
13119:Mines, Ray; Richman, Fred; Ruitenburg, Wim (1988),
13058:
11991:
7205:
6448:such that there is a polynomial equation involving
4245:
571:can help describe properties of geometric objects.
13561:, vol. 2, Nürnberg (Germany): Bauer and Raspe
13447:
13311:
13060:
12383:Groups, rings and fields: Algebra through practice
12260:(1968), "The elementary theory of finite fields",
11220:
10915:
10307:
9960:
9804:
9695:
9268:
8645:
8358:
7388:. The set of all possible orders on a fixed field
7351:
6558:
6167:A pivotal notion in the study of field extensions
5797:
5357:
5207:
3251:
3198:
3166:
2265:
2221:
1897:
1395:
13540:
11915:
11000:with field structure, which are sometimes called
10461:(i.e., the dimension is one), the function field
10457:, is invariant under birational equivalence. For
10129:, i.e., the solutions of an equation of the form
9336:(which are equipped with an exponential function
8976:if every mathematical statement that is true for
7534:Another refinement of the notion of a field is a
4897:
3975:contains a prime field. If the characteristic of
2896:
798:. Similarly, the result of the multiplication of
638:. This allows one to also consider the so-called
13696:
13229:, Springer Monographs in Mathematics, Springer,
13010:
12409:
11963:
11564:
11552:
11468:
10576:and positive characteristic, respectively. This
10094:can be performed much more efficiently than the
9972:. For example, it is an essential ingredient of
9826:of an invertible matrix leads to an isomorphism
7135:, a set-theoretic axiom that is weaker than the
6712:and its powers do not interact with elements of
4343:of this vector space is necessarily finite, say
1314:, and the multiplicative inverse (provided that
13486:Journal für die reine und angewandte Mathematik
13291:The Use of Ultraproducts in Commutative Algebra
13101:Introduction To Modern Algebra, Revised Edition
12834:
12788:Journal für die Reine und Angewandte Mathematik
12026:
10726:Local fields are completions of global fields.
10519:
10165:
9514:, which is classically defined as the group of
8154:of both fields are in bijection to one another.
8137:describing zeros of homogeneous polynomials in
7633:are filled, if there are any. For example, any
6814:(do not satisfy any polynomial relations) over
6779:. This isomorphism is obtained by substituting
5051:is built with the fractions of two elements of
3425:has characteristic 0 since no positive integer
2277:, another problem posed by the ancient Greeks.
1201:is, by definition, excluded. In order to avoid
13544:Galois Theory of Linear Differential Equations
12376:
7459:Each bounded real set has a least upper bound.
7060:, the degree of this extension is necessarily
4918:form a commutative ring, but not a field: the
3648:quite different from fields of characteristic
3167:{\displaystyle (F\smallsetminus \{0\},\cdot )}
13159:Bulletin of the American Mathematical Society
12294:Bulletin of the American Mathematical Society
12205:
10863:asks for a similarly explicit description of
10336:. In this case, one considers the algebra of
10009:Finite fields: cryptography and coding theory
7232:if any two elements can be compared, so that
5003:, the rationals, while the residue fields of
4874:
3655:
3451:and the field is said to have characteristic
3419:. For example, the field of rational numbers
3081:Additive and multiplicative groups of a field
2280:
2000:. For example, the distributive law enforces
1303:. The additive inverse of such a fraction is
579:Informally, a field is a set, along with two
524:, devoted to understanding the symmetries of
388:
13019:(2nd ed.), Cambridge University Press,
12537:, Dover Books on Mathematics Series, Dover,
12410:Borceux, Francis; Janelidze, George (2001),
12330:
11794:
11189:
11183:
10916:{\displaystyle F=\mathbf {Q} ({\sqrt {-d}})}
9682:
9670:
7956:The following topological fields are called
6394:is an algebraic element. That is to say, if
6346:A field extension in which every element of
4034:is a field with four elements. Its subfield
3246:
3240:
3193:
3187:
3152:
3146:
2287:Finite field § Field with four elements
593:, and a multiplication operation written as
555:, which is the standard general context for
12699:
12359:Beachy, John. A; Blair, William D. (2006),
12358:
11674:
11480:
11444:
11380:
11284:, p. vi) calls them "completely anomalous".
10778:, whose solutions can easily be described.
10446:, which equals the transcendence degree of
8438:, which are, by definition, those that are
7918:carries a unique norm extending the one on
7139:. In this regard, the algebraic closure of
7029:(roughly speaking, not too big compared to
6612:are rational numbers: summands of the form
1908:
13600:
13551:
13264:"Perfectoid spaces and their Applications"
12072:
9481:-adic number fields (finite extensions of
9207:, this shows that there is an isomorphism
8396:, a variant of Galois theory dealing with
7276:, which means that any quadratic equation
6092:A basic datum of a field extension is its
4785:abstractly as the rational function field
4645:. Together with a similar observation for
4398:Such a splitting field is an extension of
4207:of two strictly smaller natural numbers),
3392:If there is no positive integer such that
3304:of the multiplicative group of a field is
3099:of the field, and is sometimes denoted by
395:
381:
13288:
13221:
13171:
13072:
12596:
12306:
12003:
11809:
11642:
10318:form a field, called field of functions.
8947:
6706:. Informally speaking, the indeterminate
6442:-vector space, equals the minimal degree
6382:, as above, is an algebraic extension of
5809:Constructing fields within a bigger field
5329:
3095:under addition. This group is called the
1191:
36:Field (disambiguation) § Mathematics
13475:
12955:
12813:
12677:
12659:
12614:
12472:
12435:
12060:
12015:
11821:
11770:
11734:
11722:
11686:
11658:
11600:
11516:
11392:
11121:The a priori twofold use of the symbol "
10599:
10169:
10012:
9350:
8908:algebraically closed fields of the same
7454:
6988:algebraically closed since the equation
6108:-vector space. It satisfies the formula
4866:
4857:
4653:, also in 1770, and to a fuller extent,
4554:, who observed that permuting the zeros
4427:has as many zeros as possible since the
4349:, which implies the asserted statement.
4053:
3293:is imposed by convention to exclude the
2733:is the multiplicative identity (denoted
2078:
2074:
1912:
58:; this can be proven using the field of
42:
13565:
13541:van der Put, M.; Singer, M. F. (2003),
13314:A course in arithmetic. Translation of
13258:
13187:
12919:
12882:
12747:, Mathematical Association of America,
12735:, Universitext (2nd ed.), Springer
12493:
12161:
12143:
11904:
11857:
11845:
11646:
11588:
11576:
11540:
11528:
11504:
11492:
11281:
10736:, a global field, are the local fields
9997:systems of linear equations over a ring
8865:
6162:
4870:
3328:, it is possible to define the product
1987:, i.e., a (non-real) number satisfying
458:The best known fields are the field of
455:, and many other areas of mathematics.
14:
13697:
13574:
13483:[Algebraic Theory of Fields],
13445:
13040:
12778:
12739:
12724:
12550:
12441:Elements of the history of mathematics
12117:
11979:
11881:
11869:
11833:
11429:Mines, Richman & Ruitenburg (1988)
11268:Some authors also consider the fields
10430:The function field is invariant under
10411:indeterminates. The function field of
10355:function field of an algebraic variety
9884:Linear algebra and commutative algebra
8369:
8106:statement that is true for almost all
7167:). For any algebraically closed field
6649:
6590:consisting of all numbers of the form
5491:ideal generated by a single polynomial
5431:. Any field obtained in this way is a
5374:is the field of fractions of the ring
4892:
4847:
4058:In modular arithmetic modulo 12,
13637:
13400:
13371:
13342:
13306:
13152:
13098:
12922:Archive for History of Exact Sciences
12821:, vol. 1 (2nd ed.), Dover,
12567:
12530:
12179:
12095:
12083:
12048:
12037:
11893:
11782:
11758:
11746:
11710:
11670:
11612:
11456:
11416:
11404:
11245:Further examples include the maximal
10730:asserts that the only completions of
10308:{\displaystyle {\frac {f(x)}{g(x)}},}
7977:(local fields of characteristic zero)
7529:
7177:, the algebraic closure of the field
7105:. For example, the algebraic closure
6868:
6751:of a polynomial with coefficients in
6735:discussed above is a key example: if
6702:, which are not algebraic are called
5236:over a field (or an integral domain)
5023:
4824:
4747:
4167:results in the above-mentioned field
3252:{\displaystyle F\smallsetminus \{0\}}
3199:{\displaystyle F\smallsetminus \{0\}}
2860:
708:
56:straightedge and compass construction
13528:
12981:
12835:Jannsen, Uwe; Wingberg, Kay (1982),
12745:A Guide to Groups, Rings, and Fields
12287:
12128:
12106:
11951:
11939:
11927:
11698:
10853:obtained by adjoining all primitive
10427:by a (slightly) smaller subvariety.
10321:This occurs in two main cases. When
9991:(the analogue of vector spaces over
8575:fundamental theorem of Galois theory
7712:, i.e., a sequence whose limit (for
6400:is algebraic, all other elements of
4944:in which every nonzero element is a
4193:which can be expressed as a product
3227:is an abelian group under addition,
2986:. This means that every field is an
2798:, as required by the distributivity.
975:: there exist two distinct elements
11992:Marker, Messmer & Pillay (2006)
8944:isomorphic as topological fields).
8876:include the characteristic and the
8519:For a finite Galois extension, the
7996:, the field of Laurent series over
6678:, is not an algebraic extension of
6021:
4808:(1882) proved the transcendence of
4362:elements can be constructed as the
3636:is compatible with the addition in
3215:A field may thus be defined as set
2810:with four elements, and is denoted
2721:. The notation is chosen such that
1256:
1176:Even more succinctly: a field is a
528:, provides an elegant proof of the
54:cannot be constructed using only a
24:
13227:The theory of classical valuations
12680:A First Course In Abstract Algebra
12256:
11773:, Chapter V, §14, No. 2, Theorem 1
10947:
9498:and of related groups such as the
9228:
8337:
8258:
8067:. (However, since the addition in
6332:, since it satisfies the equation
5843:, there is a smallest subfield of
5291:
4862:
3311:Root of unity § Cyclic groups
501:, i.e., fields with finitely many
25:
13726:
13690:
13606:Introduction to Cyclotomic Fields
13481:"Algebraische Theorie der Körper"
13379:, Jones and Bartlett Publishers,
11019:
9376:. By elementary means, the group
9049:The set of such formulas for all
7213:
7196:, obtained by adjoining roots of
5922:, there is a minimal subfield of
5401:
5242:is the field of fractions of the
4928:is not itself an integer, unless
4873:can be found in Steinitz's work.
4114:" means to work with the numbers
3317:
1126:of multiplication over addition:
760:a uniquely determined element of
508:The theory of fields proves that
13553:von Staudt, Karl Georg Christian
13190:Lectures on formally real fields
13121:A course in constructive algebra
11148:⟨2, 2, 1, 1, 0, 0⟩
10893:
10815:, describes the maximal abelian
10701:used cyclotomic fields to prove
10604:The fifth roots of unity form a
9708:norm residue isomorphism theorem
9496:Representations of Galois groups
9259:
9240:
8403:
8290:
8226:
8133:. An application of this is the
8007:(local fields of characteristic
7671:, in the sense that distance of
7206:Fields with additional structure
7080:. Such fields are also known as
6863:purely transcendental extensions
6721:Once again, the field extension
5788:
5721:
4711:(polynomial equations of degree
4160:. For example, taking the prime
4007:
1331:, which can be seen as follows:
929:of addition and multiplication:
871:of addition and multiplication:
764:. The result of the addition of
611:, including the existence of an
443:. A field is thus a fundamental
13570:, SUMS, vol. 151, Springer
13173:10.1090/S0002-9904-1893-00178-X
12886:(2007), Kleiner, Israel (ed.),
12598:Dirichlet, Peter Gustav Lejeune
12333:Z. Math. Logik Grundlagen Math.
12122:
12111:
12100:
12089:
12077:
12066:
12054:
12042:
12031:
12020:
12009:
11997:
11985:
11973:
11957:
11945:
11933:
11921:
11916:van der Put & Singer (2003)
11909:
11898:
11887:
11875:
11863:
11851:
11839:
11827:
11824:, Chapter VI, §2.3, Corollary 1
11815:
11803:
11788:
11776:
11764:
11752:
11740:
11728:
11716:
11704:
11692:
11680:
11664:
11652:
11636:
11618:
11606:
11594:
11582:
11570:
11558:
11546:
11534:
11522:
11510:
11498:
11486:
11474:
11383:, Definition 4.1.1, p. 181
11351:
11337:and finite field extensions of
11321:
11287:
11262:
11239:
11128:
10154:Finite fields are also used in
9878:
9370:finite separable extensions of
8754:is regarded as a polynomial in
8577:, which constructs an explicit
7951:
6980:is algebraically closed, i.e.,
5912:. More generally, for a subset
5831:as a subfield. For any element
5595:) which satisfies the equation
4703:. Building on Lagrange's work,
4102:. For a fixed positive integer
2871:denotes an arbitrary field and
13454:, Cambridge University Press,
12414:, Cambridge University Press,
12363:(3 ed.), Waveland Press,
11964:Borceux & Janelidze (2001)
11565:Lidl & Niederreiter (2008)
11553:Lidl & Niederreiter (2008)
11469:Lidl & Niederreiter (2008)
11462:
11450:
11438:
11422:
11410:
11398:
11386:
11374:
11115:
10910:
10897:
10417:is the same as the one of any
10296:
10290:
10282:
10276:
9796:
9769:
9745:
9739:
9652:
9640:
9589:
9583:
9225:
8858:This fact is the beginning of
8312:
8309:
8303:
8300:
7655:, is a "gap" in the rationals
7097:algebraic closure and denoted
6972:fundamental theorem of algebra
6454:, as above. If this degree is
5772:
5731:
5725:
5352:
5319:
4985:, i.e., two ways of modifying
4898:Constructing fields from rings
4261:from being a field. The field
3932:. All field homomorphisms are
3161:
3137:
2897:Consequences of the definition
1707:
1689:
13:
1:
12888:A history of abstract algebra
12624:Graduate Texts in Mathematics
12317:10.1090/S0273-0979-01-00934-X
12162:Allenby, R. B. J. T. (1991),
12136:
12027:Jannsen & Wingberg (1982)
11016:, form such a Field as well.
10239:This makes these functions a
10174:A compact Riemann surface of
9152:is a field. It is denoted by
9061:is algebraically closed. The
8398:linear differential equations
7896:is used in number theory and
6826:is an algebraic extension of
6654:The above-mentioned field of
6215:, that is, if it satisfies a
4831:'s notion included the field
4663:(1801), studied the equation
4329:. This statement holds since
3805:between two fields such that
3601:-fold product of the element
3457:then. For example, the field
3431:is zero. Otherwise, if there
2222:{\displaystyle h={\sqrt {p}}}
1184:and all nonzero elements are
1080:, there exists an element in
1027:, there exists an element in
574:
470:. Many other fields, such as
12575:(2nd ed.), Birkhäuser,
12534:Elements of Abstract Algebra
12146:Introduction to Field Theory
11368:
11327:More precisely, there is an
11134:Equivalently, a field is an
11053:(which is a field), and the
10585:concerning the zeros of the
10520:Number theory: global fields
10440:abstract algebraic varieties
10166:Geometry: field of functions
9244:
9027:= "any polynomial of degree
8870:Basic invariants of a field
8860:Grothendieck's Galois theory
8117:is also true for almost all
7608:between any two elements of
7394:is isomorphic to the set of
6865:) and algebraic extensions.
5979:is the smallest subfield of
5514:) is maximal if and only if
4912:. For example, the integers
4639:to a quadratic equation for
4183:and more generally, for any
472:fields of rational functions
7:
13188:Prestel, Alexander (1984),
12767:Encyclopedia of Mathematics
12626:, vol. 150, New York:
11099:Wedderburn's little theorem
11004:s, with a capital 'F'. The
10535:. They are, by definition,
10123:elliptic curve cryptography
9818:is related to the group of
9550:
8418:of a field by studying the
8377:are fields equipped with a
8082:, which is not the case in
7044:is an algebraic closure of
5476:, this field is called the
4875:Artin & Schreier (1927)
4660:Disquisitiones Arithmeticae
4490:has only two zeros (namely
3445:. It is usually denoted by
3280:are uniquely determined by
3107:when denoting it simply as
2199:, at a distance of exactly
1251:
642:operations of subtraction,
10:
13731:
13566:Wallace, D. A. R. (1998),
13123:, Universitext, Springer,
12958:Ordered exponential fields
12678:Fraleigh, John B. (1976),
12387:Cambridge University Press
10877:imaginary quadratic fields
10545:) or function fields over
10083:in a (large) finite field
9976:and of the proof that any
9961:{\displaystyle x=a^{-1}b.}
8778:, for some indeterminates
8646:{\displaystyle {\sqrt{~}}}
8407:
8394:differential Galois theory
8152:Tamely ramified extensions
7217:
6159:is an infinite extension.
6025:
5469:has only one maximal ideal
4850:to introduce the field of
4537:
4339:over its prime field. The
4244:, which, as was explained
4011:
3926:are arbitrary elements of
3656:Subfields and prime fields
2284:
2281:A field with four elements
2266:{\displaystyle {\sqrt{2}}}
2113:
1920:
1260:
1209:operations (the constants
634:for every nonzero element
29:
13614:10.1007/978-1-4612-1934-7
13568:Groups, Rings, and Fields
13499:10.1515/crll.1910.137.167
13235:10.1007/978-1-4612-0551-7
13129:10.1007/978-1-4419-8640-5
12995:10.1007/978-1-4613-0041-0
12896:10.1007/978-0-8176-4685-1
12682:(2nd ed.), Reading:
12636:10.1007/978-1-4612-5350-1
12449:10.1007/978-3-642-61693-8
12390:. See especially Book 3 (
12291:(2002), "The octonions",
12166:, Butterworth-Heinemann,
11675:Fricke & Weber (1924)
11481:Beachy & Blair (2006)
11445:Beachy & Blair (2006)
11381:Beachy & Blair (2006)
11329:equivalence of categories
10869:of general number fields
10652:, i.e., a complex number
8579:one-to-one correspondence
8448:primitive element theorem
7056:is quite special: by the
6812:algebraically independent
6425:, i.e., the dimension of
6096:, i.e., the dimension of
6032:The notion of a subfield
5861:, called the subfield of
5127:are equal if and only if
4973:Given a commutative ring
4887:primitive element theorem
4769:defined what he called a
4144:of integers, dividing by
2498:
2303:
2298:
2295:
810:is called the product of
476:algebraic function fields
13289:Schoutens, Hans (2002),
12956:Kuhlmann, Salma (2000),
12660:Escofier, J. P. (2012),
12507:10.1017/CBO9781139171885
12478:Algebra II. Chapters 4–7
12345:10.1002/malq.19920380136
12164:Rings, Fields and Groups
11645:, p. 42, translation by
11555:, Lemma 2.1, Theorem 2.2
11228:are abelian groups, and
11108:
10527:are in the limelight in
10072:factors, for an integer
10017:The sum of three points
9710:, proved around 2000 by
8808:tensor product of fields
7905:. The algebraic closure
6741:is not algebraic (i.e.,
6376:generated by an element
6028:Glossary of field theory
4409:in which the polynomial
3962:are called isomorphic).
3382:(which is an element of
3338:of an arbitrary element
2183:), which intersects the
2122:compass and straightedge
1909:Real and complex numbers
1219:constructive mathematics
738:. A binary operation on
518:compass and straightedge
447:which is widely used in
13602:Washington, Lawrence C.
13450:Rings and factorization
13377:Topics in Galois theory
13099:McCoy, Neal H. (1968),
12202:, especially Chapter 13
12144:Adamson, I. T. (2007),
11968:Étale fundamental group
11101:states that all finite
10861:Kronecker's Jugendtraum
10813:Kronecker–Weber theorem
10759:Hasse–Minkowski theorem
10529:algebraic number theory
10492:(the analogue of being
9389:can be shown to be the
8974:elementarily equivalent
7119:is called the field of
7072:elementarily equivalent
6791:in rational fractions.
6569:For example, the field
6460:, then the elements of
6054:(or just extension) of
5498:in the polynomial ring
4806:Ferdinand von Lindemann
4800:'s work in 1844, until
4686:. Gauss deduced that a
4544:algebraic number theory
4469:, so they are zeros of
4090:is not a field because
2802:This field is called a
1949:consist of expressions
1203:existential quantifiers
1065:Multiplicative inverses
973:multiplicative identity
713:Formally, a field is a
495:cryptographic protocols
480:algebraic number fields
13446:Sharpe, David (1987),
13063:Model theory of fields
13041:Lorenz, Falko (2008),
12705:Weber, Heinrich Martin
12148:, Dover Publications,
11222:
10958:field with one element
10932:complex multiplication
10917:
10797:for some number field
10755:local–global principle
10609:
10578:function field analogy
10556:(finite extensions of
10539:(finite extensions of
10436:birational equivalence
10309:
10179:
10042:
9962:
9914:has a unique solution
9806:
9697:
9270:
8948:Model theory of fields
8647:
8611:(cannot be built from
8414:Galois theory studies
8360:
7460:
7362:only has the solution
7353:
7270:Artin–Schreier theorem
7058:Artin–Schreier theorem
6837:. Any field extension
6560:
6507:
5799:
5710:yields an isomorphism
5546:. This yields a field
5359:
5295:
5209:
5009:are the finite fields
4773:, which is a field of
4763:
4760:Richard Dedekind, 1871
4729:algebraic number field
4696:can be constructed if
4095:
4094:is not a prime number.
4076:leaves remainder
3253:
3200:
3168:
3085:The axioms of a field
2267:
2223:
2111:
2085:geometric mean theorem
1918:
1899:
1397:
1192:Alternative definition
1188:under multiplication.
1101:multiplicative inverse
627:multiplicative inverse
516:cannot be done with a
63:
34:. For other uses, see
13647:Mathematische Annalen
13575:Warner, Seth (1989),
12713:(in German), Vieweg,
11232:is distributive over
11223:
10918:
10783:Inverse Galois theory
10703:Fermat's Last Theorem
10603:
10587:Riemann zeta function
10514:minimal model program
10342:meromorphic functions
10338:holomorphic functions
10310:
10173:
10035:on an elliptic curve
10016:
9963:
9807:
9698:
9357:absolute Galois group
9351:Absolute Galois group
9271:
9119:is a field for every
8992:sentences (involving
8648:
8615:), then the zeros of
8361:
7980:finite extensions of
7966:finite extensions of
7795:, for a prime number
7524:non-standard analysis
7458:
7354:
6632:can be simplified to
6561:
6481:
6026:Further information:
5800:
5360:
5275:
5210:
4877:linked the notion of
4829:Heinrich Martin Weber
4771:domain of rationality
4752:
4647:equations of degree 4
4552:Joseph-Louis Lagrange
4515:elements, denoted by
4459:satisfy the equation
4108:, arithmetic "modulo
4057:
3551:binomial coefficients
3254:
3201:
3169:
2268:
2224:
2126:constructible numbers
2116:Constructible numbers
2082:
2075:Constructible numbers
2068:Cartesian coordinates
1916:
1900:
1398:
776:is called the sum of
545:mathematical analysis
60:constructible numbers
46:
13710:Algebraic structures
13412:, Berlin, New York:
13316:Cours d'arithmetique
13013:Niederreiter, Harald
12710:Lehrbuch der Algebra
12557:On Numbers and Games
11247:unramified extension
11169:
10883:
10267:
9974:Gaussian elimination
9930:
9721:
9565:
9395:profinite completion
9214:
8940:are isomorphic (but
8878:transcendence degree
8866:Invariants of fields
8674:Abel–Ruffini theorem
8665:is not solvable for
8628:
8544:that are trivial on
8454:, i.e., of the form
8416:algebraic extensions
8210:
8157:Adjoining arbitrary
7463:An ordered field is
7283:
6929:, with coefficients
6875:algebraically closed
6696:. Elements, such as
6478:
6163:Algebraic extensions
5717:
5675:is irreducible over
5579:contains an element
5272:
5144:
4879:orderings in a field
4707:claimed (1799) that
4655:Carl Friedrich Gauss
3965:A field is called a
3595:th power, i.e., the
3519:. This implies that
3231:
3178:
3134:
3128:multiplicative group
3113:could be confusing.
3091:imply that it is an
2248:
2203:
1413:
1338:
1165:under addition with
530:Abel-Ruffini theorem
292:Group with operators
235:Complemented lattice
70:Algebraic structures
13705:Field (mathematics)
12853:1982InMat..70...71J
12741:Gouvêa, Fernando Q.
12726:Gouvêa, Fernando Q.
12552:Conway, John Horton
11795:Banaschewski (1992)
11136:algebraic structure
11041:-vector spaces are
10859:th roots of unity.
10757:. For example, the
10728:Ostrowski's theorem
10593:(proven in 1974 by
10533:arithmetic geometry
10510:birational geometry
10247:commutative algebra
9845:Matsumoto's theorem
9820:invertible matrices
9795:
9738:
9582:
9510:. For example, the
9063:Lefschetz principle
8653:. For example, the
8581:between the set of
8536:field automorphisms
8512:has characteristic
8375:Differential fields
8370:Differential fields
8180:), respectively of
7854:obtained using the
7788:obtained using the
7538:, in which the set
7342:
7318:
7300:
7274:formally real field
7161:(the ones of order
6879:polynomial equation
6808:transcendence basis
6650:Transcendence bases
6584:is the subfield of
6360:algebraic extension
6305:. For example, the
6217:polynomial equation
5882:. The passage from
5383:formal power series
4893:Constructing fields
4867:Constructing fields
4335:may be viewed as a
4300:Every finite field
4072:, which divided by
3789:Field homomorphisms
3752:, and that for all
3490:has characteristic
3466:has characteristic
3435:a positive integer
2292:
514:squaring the circle
445:algebraic structure
346:Composition algebra
106:Quasigroup and loop
13659:10.1007/BF01446451
13577:Topological fields
13402:Serre, Jean-Pierre
13373:Serre, Jean-Pierre
13344:Serre, Jean-Pierre
13308:Serre, Jean-Pierre
13198:10.1007/BFb0101548
13154:Moore, E. Hastings
12934:10.1007/BF00327219
12861:10.1007/bf01393199
12531:Clark, A. (1984),
12227:10.1007/BF02952522
11848:, Proposition 1.22
11761:, Corollary 13.3.6
11673:, p. 33. See also
11507:, §I.2, p. 10
11431:, §II.2. See also
11359:commutative fields
11218:
11143:, +, ⋅, −, , 0, 1⟩
11081:algebraic topology
10913:
10827:: it is the field
10809:abelian extensions
10805:Class field theory
10610:
10583:Riemann hypothesis
10305:
10180:
10096:discrete logarithm
10043:
9958:
9816:Algebraic K-theory
9802:
9778:
9724:
9712:Vladimir Voevodsky
9693:
9568:
9334:exponential fields
9330:real closed fields
9266:
8958:mathematical logic
8795:is any field, and
8643:
8356:
7937:and is denoted by
7530:Topological fields
7494:1/2, 1/3, 1/4, ...
7461:
7396:ring homomorphisms
7349:
7328:
7304:
7286:
7173:of characteristic
7082:real closed fields
6869:Closure operations
6656:rational fractions
6582:Gaussian rationals
6556:
6352:is algebraic over
6320:is algebraic over
6179:algebraic elements
5899:is referred to by
5795:
5639:, which satisfies
5355:
5234:rational fractions
5205:
5072:are the fractions
5039:field of fractions
5024:Field of fractions
4871:Elementary notions
4775:rational fractions
4627:Scipione del Ferro
4581:in the expression
4548:algebraic geometry
4509:finite field with
4475:. By contrast, in
4421:zeros. This means
4187:(i.e., any number
4100:modular arithmetic
4096:
3946:, it is called an
3249:
3196:
3164:
2861:Elementary notions
2291:
2275:cube with volume 2
2263:
2219:
2112:
1919:
1895:
1893:
1393:
844:(in these axioms,
722:together with two
709:Classic definition
538:solved in radicals
491:algebraic geometry
64:
13579:, North-Holland,
13423:978-3-540-42192-4
13406:Galois cohomology
13300:978-3-642-13367-1
13282:978-89-6105-804-9
13275:, Kyung Moon SA,
13084:978-1-56881-282-3
13052:978-0-387-72487-4
13026:978-0-521-06567-2
12905:978-0-8176-4684-4
12828:978-0-486-47189-1
12754:978-0-88385-355-9
12671:978-1-4613-0191-2
12602:Dedekind, Richard
12544:978-0-486-64725-8
12495:Cassels, J. W. S.
12474:Bourbaki, Nicolas
12437:Bourbaki, Nicolas
12198:978-0-13-004763-2
12173:978-0-340-54440-2
12155:978-0-486-46266-0
12073:Washington (1997)
11799:Mathoverflow post
11251:abelian extension
11162:, +, −, 0⟩
11064:Frobenius theorem
11012:, a concept from
10908:
10613:Cyclotomic fields
10472:is very close to
10364:regular functions
10300:
10187:topological space
10005:of the integers.
9508:Galois cohomology
9504:Langlands program
9247:
8982:is also true for
8641:
8635:
8494:are contained in
8427:Galois extensions
8204:perfectoid fields
8135:Ax–Kochen theorem
7883:
7882:
7635:irrational number
7546:topological space
7536:topological field
7465:Dedekind-complete
7430:Archimedean field
7133:ultrafilter lemma
7121:algebraic numbers
7010:algebraic closure
6768:is isomorphic to
6536:
6533:
5958:of two subfields
5786:
5781:
5767:
5621:is obtained from
5318:
5200:
5168:
5155:
4844:Giuseppe Veronese
4827:. In particular,
4767:Leopold Kronecker
4717:Niels Henrik Abel
4709:quintic equations
3607:. Therefore, the
3557:are divisible by
3553:appearing in the
3130:, and denoted by
2865:In this section,
2695:
2694:
2691:
2690:
2496:
2495:
2261:
2217:
1886:
1873:
1860:
1847:
1826:
1803:
1780:
1751:
1722:
1670:
1638:
1620:
1597:
1569:
1543:
1530:
1517:
1504:
1486:
1460:
1447:
1429:
1385:
1362:
1349:
1013:Additive inverses
822:, and is denoted
788:, and is denoted
724:binary operations
621:for all elements
534:quintic equations
466:and the field of
405:
404:
16:(Redirected from
13722:
13715:Abstract algebra
13685:
13634:
13597:
13571:
13562:
13548:
13537:
13525:
13493:(137): 167–309,
13472:
13453:
13442:
13397:
13368:
13339:
13320:
13303:
13285:
13268:
13255:
13223:Ribenboim, Paulo
13218:
13184:
13175:
13149:
13115:
13095:
13076:
13066:
13055:
13037:
13007:
12978:
12952:
12916:
12879:
12831:
12815:Jacobson, Nathan
12810:
12775:
12757:
12736:
12721:
12696:
12674:
12656:
12611:
12593:
12564:
12547:
12527:
12490:
12469:
12432:
12389:
12379:Robertson, E. F.
12373:
12361:Abstract Algebra
12355:
12327:
12310:
12284:
12253:
12201:
12176:
12158:
12131:
12126:
12120:
12115:
12109:
12104:
12098:
12093:
12087:
12081:
12075:
12070:
12064:
12063:, §13, Theorem A
12058:
12052:
12046:
12040:
12035:
12029:
12024:
12018:
12013:
12007:
12004:Schoutens (2002)
12001:
11995:
11989:
11983:
11977:
11971:
11961:
11955:
11954:, Example VI.2.6
11949:
11943:
11937:
11931:
11925:
11919:
11913:
11907:
11902:
11896:
11891:
11885:
11879:
11873:
11867:
11861:
11855:
11849:
11843:
11837:
11836:, §22, Theorem 1
11831:
11825:
11819:
11813:
11810:Ribenboim (1999)
11807:
11801:
11792:
11786:
11780:
11774:
11768:
11762:
11756:
11750:
11749:, Theorem 13.3.4
11744:
11738:
11732:
11726:
11720:
11714:
11708:
11702:
11696:
11690:
11684:
11678:
11668:
11662:
11656:
11650:
11643:Dirichlet (1871)
11640:
11634:
11633:
11622:
11616:
11610:
11604:
11598:
11592:
11586:
11580:
11574:
11568:
11562:
11556:
11550:
11544:
11538:
11532:
11526:
11520:
11514:
11508:
11502:
11496:
11490:
11484:
11478:
11472:
11466:
11460:
11454:
11448:
11442:
11436:
11426:
11420:
11414:
11408:
11402:
11396:
11390:
11384:
11378:
11362:
11355:
11349:
11347:
11336:
11325:
11319:
11317:
11311:
11298:
11291:
11285:
11279:
11273:
11266:
11260:
11258:
11243:
11237:
11235:
11231:
11227:
11225:
11224:
11219:
11217:
11213:
11212:
11211:
11203:
11164:
11163:
11155:is not defined,
11154:
11150:
11149:
11144:
11132:
11126:
11124:
11119:
11074:
11061:
11052:
11046:
11040:
10980:
10976:
10970:
10955:
10939:
10930:, the theory of
10929:
10922:
10920:
10919:
10914:
10909:
10901:
10896:
10874:
10868:
10858:
10849:
10826:
10820:
10802:
10796:
10777:
10766:
10752:
10746:
10735:
10721:
10692:
10686:
10675:
10673:
10666:
10664:
10657:
10647:
10641:
10630:
10606:regular pentagon
10591:Weil conjectures
10575:
10571:
10555:
10544:
10507:
10501:
10483:
10477:
10471:
10456:
10426:
10416:
10410:
10404:
10381:
10371:
10361:
10349:
10335:
10329:complex manifold
10326:
10314:
10312:
10311:
10306:
10301:
10299:
10285:
10271:
10258:integral domains
10244:
10234:
10233:
10197:
10193:
10149:
10116:
10103:
10093:
10078:
10071:
10065:
10040:
10034:
10028:
10022:
10004:
9967:
9965:
9964:
9959:
9951:
9950:
9925:
9919:
9910:
9894:
9874:
9860:
9842:
9811:
9809:
9808:
9803:
9794:
9786:
9768:
9767:
9752:
9737:
9732:
9702:
9700:
9699:
9694:
9689:
9685:
9628:
9623:
9622:
9604:
9603:
9581:
9576:
9545:
9521:
9491:
9480:
9470:
9435:
9428:
9415:
9402:
9388:
9375:
9365:
9346:
9323:
9284:
9275:
9273:
9272:
9267:
9262:
9254:
9253:
9248:
9243:
9238:
9232:
9231:
9206:
9202:
9187:
9172:
9151:
9146:with respect to
9145:
9130:
9124:
9118:
9107:
9097:
9088:
9082:
9076:
9070:
9060:
9055:expresses that
9054:
9044:
9038:
9032:
9026:
9012:
9006:
8999:
8995:
8987:
8981:
8971:
8965:
8939:
8933:
8918:
8903:
8897:
8891:
8885:
8875:
8853:
8831:
8825:
8819:
8801:
8794:
8788:
8777:
8753:
8747:
8739:
8724:
8707:
8697:
8671:
8664:
8655:symmetric groups
8652:
8650:
8649:
8644:
8642:
8640:
8633:
8632:
8620:
8606:
8596:
8572:
8566:
8549:
8543:
8534:is the group of
8533:
8515:
8511:
8505:
8499:
8493:
8487:
8477:
8471:
8437:
8391:
8365:
8363:
8362:
8357:
8352:
8348:
8347:
8343:
8342:
8341:
8340:
8331:
8299:
8298:
8293:
8273:
8269:
8268:
8264:
8263:
8262:
8261:
8252:
8235:
8234:
8229:
8201:
8185:
8179:
8168:
8163:-power roots of
8162:
8147:
8132:
8116:
8097:
8077:
8066:
8052:(referred to as
8051:
8031:
8012:
8006:
7995:
7976:
7947:
7928:
7917:
7911:
7902:
7895:
7879:
7872:
7860:-adic valuation
7859:
7850:
7843:
7829:
7819:
7812:
7800:
7785:
7776:
7758:
7756:
7743:
7722:
7721:
7718:
7707:
7705:
7686:
7676:
7670:
7660:
7654:
7653:
7652:
7641:
7632:
7626:
7613:
7604:that measures a
7600:
7567:
7557:
7543:
7520:
7504:
7495:
7490:
7451:
7438:
7424:
7418:
7408:
7393:
7387:
7358:
7356:
7355:
7350:
7341:
7336:
7317:
7312:
7299:
7294:
7267:
7263:
7256:
7249:
7242:
7201:
7192:is the field of
7187:
7176:
7172:
7166:
7160:
7149:
7130:
7129:
7118:
7112:
7111:
7104:
7103:
7092:
7079:
7069:
7063:
7055:
7049:
7043:
7034:
7028:
7018:
7006:
6997:
6979:
6969:
6955:
6928:
6860:
6846:
6836:
6825:
6819:
6805:
6799:
6790:
6784:
6778:
6767:
6756:
6746:
6740:
6734:
6717:
6711:
6701:
6695:
6689:
6683:
6673:
6667:
6645:
6631:
6617:
6611:
6605:
6599:
6589:
6579:
6565:
6563:
6562:
6557:
6546:
6545:
6534:
6531:
6527:
6526:
6517:
6516:
6506:
6495:
6470:
6459:
6453:
6447:
6441:
6435:
6424:
6410:
6399:
6393:
6387:
6381:
6375:
6357:
6351:
6341:
6331:
6326:, and even over
6325:
6319:
6313:
6304:
6292:
6288:
6266:
6214:
6198:
6190:
6176:
6158:
6148:
6144:
6128:
6114:
6107:
6101:
6088:
6082:
6072:
6059:
6047:
6041:
6022:Field extensions
6017:
6011:
6005:
5999:
5998:
5990:
5985:containing both
5984:
5978:
5972:
5971:
5963:
5950:
5939:
5933:
5927:
5921:
5911:
5898:
5887:
5881:
5870:
5860:
5854:
5848:
5842:
5836:
5830:
5824:
5818:
5804:
5802:
5801:
5796:
5791:
5784:
5783:
5782:
5780:
5775:
5770:
5765:
5764:
5760:
5753:
5752:
5738:
5724:
5709:
5703:
5696:
5687:
5680:
5674:
5668:
5658:
5651:
5645:
5638:
5626:
5620:
5610:
5604:
5594:
5584:
5578:
5569:
5563:
5541:
5535:
5529:
5519:
5513:
5507:
5497:
5485:
5475:
5467:
5461:
5451:
5445:
5444:
5430:
5420:
5411:
5397:
5391:
5380:
5373:
5364:
5362:
5361:
5356:
5345:
5344:
5332:
5316:
5315:
5314:
5305:
5304:
5294:
5289:
5261:
5250:
5241:
5231:
5214:
5212:
5211:
5206:
5201:
5199:
5191:
5174:
5169:
5161:
5156:
5148:
5136:
5126:
5116:
5107:. Two fractions
5106:
5099:
5093:
5087:
5081:
5071:
5056:
5050:
5036:
5019:
5008:
5002:
4996:
4990:
4984:
4978:
4965:
4961:
4955:
4943:
4934:
4927:
4917:
4911:
4904:commutative ring
4855:
4841:
4820:, respectively.
4819:
4813:
4798:Joseph Liouville
4795:
4784:
4761:
4736:Richard Dedekind
4714:
4702:
4693:
4681:
4672:
4644:
4638:
4620:
4611:
4579:cubic polynomial
4576:
4533:
4525:
4514:
4503:
4497:
4493:
4489:
4483:
4474:
4468:
4458:
4449:
4442:
4436:
4426:
4420:
4414:
4408:
4393:
4388:
4378:
4361:
4348:
4334:
4328:
4321:
4316:elements, where
4315:
4305:
4296:
4285:
4279:
4273:
4260:
4243:
4230:
4219:
4206:
4192:
4185:composite number
4182:
4175:
4166:
4155:
4149:
4143:
4134:
4113:
4107:
4093:
4089:
4079:
4075:
4071:
4065:
4061:
4050:
4046:
4042:
4033:
4003:
3997:
3986:
3980:
3974:
3961:
3955:
3941:
3931:
3925:
3916:
3907:
3889:
3847:
3804:
3784:
3778:
3771:
3764:
3758:
3751:
3745:
3735:
3725:
3711:
3704:
3700:
3694:
3688:
3682:
3676:
3670:
3651:
3647:
3641:
3632:
3631:
3606:
3600:
3594:
3589:factors) is the
3588:
3582:
3569:
3562:
3555:binomial formula
3549:since all other
3544:
3543:
3537:
3518:
3512:
3506:
3495:
3489:
3480:
3469:
3465:
3456:
3450:
3440:
3430:
3424:
3418:
3412:is said to have
3411:
3401:
3387:
3381:
3364:
3358:
3349:
3343:
3337:
3327:
3292:
3289:The requirement
3285:
3279:
3273:
3258:
3256:
3255:
3250:
3226:
3220:
3211:
3205:
3203:
3202:
3197:
3173:
3171:
3170:
3165:
3125:
3112:
3106:
3090:
3076:
3069:
3057:
3028:
3016:
3011:
3004:
2998:
2985:
2955:
2948:
2944:
2938:
2932:
2922:
2918:
2907:
2892:
2882:
2876:
2870:
2856:
2852:
2838:
2832:
2822:
2818:
2797:
2796:
2765:
2736:
2732:
2726:
2720:
2714:
2708:
2702:
2687:
2680:
2673:
2666:
2659:
2650:
2643:
2636:
2629:
2622:
2613:
2606:
2599:
2598:
2590:
2589:
2581:
2572:
2565:
2558:
2557:
2549:
2548:
2540:
2531:
2524:
2517:
2510:
2500:
2499:
2492:
2485:
2478:
2471:
2464:
2455:
2448:
2441:
2434:
2427:
2418:
2411:
2404:
2403:
2395:
2394:
2386:
2377:
2370:
2363:
2362:
2354:
2353:
2345:
2336:
2329:
2322:
2315:
2305:
2304:
2293:
2290:
2272:
2270:
2269:
2264:
2262:
2260:
2252:
2241:has length one.
2240:
2234:
2228:
2226:
2225:
2220:
2218:
2213:
2198:
2192:
2182:
2173:
2163:
2157:
2151:
2141:
2109:
2103:
2096:
2058:
1999:
1993:
1982:
1972:
1962:
1948:
1939:
1904:
1902:
1901:
1896:
1894:
1887:
1879:
1874:
1866:
1861:
1853:
1848:
1840:
1836:
1827:
1825:
1817:
1809:
1804:
1802:
1794:
1786:
1781:
1779:
1768:
1757:
1752:
1750:
1739:
1728:
1723:
1721:
1710:
1684:
1680:
1671:
1669:
1661:
1644:
1639:
1631:
1626:
1622:
1621:
1619:
1611:
1603:
1598:
1596:
1588:
1580:
1570:
1562:
1558:
1549:
1545:
1544:
1536:
1531:
1523:
1518:
1510:
1505:
1497:
1487:
1479:
1475:
1466:
1462:
1461:
1453:
1448:
1440:
1430:
1422:
1419:
1402:
1400:
1399:
1394:
1386:
1384:
1376:
1368:
1363:
1355:
1350:
1342:
1330:
1320:
1313:
1302:
1291:
1285:
1279:
1257:Rational numbers
1247:
1236:
1232:
1228:
1216:
1212:
1199:Division by zero
1183:
1178:commutative ring
1172:
1168:
1156:
1119:
1108:
1098:
1091:
1085:
1079:
1073:
1060:
1049:
1042:additive inverse
1039:
1032:
1026:
1020:
1008:
998:
988:
982:
978:
964:
946:
922:
896:
863:
855:
851:
847:
837:
827:
821:
815:
809:
803:
797:
787:
781:
775:
769:
763:
759:
755:
741:
729:
721:
703:
683:
661:
652:, and division,
651:
637:
633:
624:
620:
613:additive inverse
605:rational numbers
602:
592:
526:field extensions
510:angle trisection
460:rational numbers
397:
390:
383:
172:Commutative ring
101:Rack and quandle
66:
65:
21:
13730:
13729:
13725:
13724:
13723:
13721:
13720:
13719:
13695:
13694:
13693:
13688:
13639:Weber, Heinrich
13624:
13587:
13477:Steinitz, Ernst
13462:
13424:
13414:Springer-Verlag
13387:
13358:
13329:
13301:
13283:
13266:
13245:
13208:
13139:
13105:Allyn and Bacon
13085:
13053:
13027:
13005:
12968:
12928:(1–2): 40–154,
12906:
12884:Kleiner, Israel
12829:
12760:
12755:
12694:
12672:
12646:
12628:Springer-Verlag
12616:Eisenbud, David
12583:
12545:
12517:
12488:
12459:
12422:
12412:Galois theories
12371:
12274:10.2307/1970573
12199:
12174:
12156:
12139:
12134:
12127:
12123:
12116:
12112:
12105:
12101:
12094:
12090:
12082:
12078:
12071:
12067:
12061:Eisenbud (1995)
12059:
12055:
12047:
12043:
12036:
12032:
12025:
12021:
12016:Kuhlmann (2000)
12014:
12010:
12002:
11998:
11994:, Corollary 1.2
11990:
11986:
11982:, Theorem 6.4.8
11978:
11974:
11962:
11958:
11950:
11946:
11938:
11934:
11930:, Theorem V.4.6
11926:
11922:
11914:
11910:
11903:
11899:
11892:
11888:
11880:
11876:
11868:
11864:
11856:
11852:
11844:
11840:
11832:
11828:
11822:Bourbaki (1988)
11820:
11816:
11808:
11804:
11793:
11789:
11781:
11777:
11771:Bourbaki (1988)
11769:
11765:
11757:
11753:
11745:
11741:
11735:Jacobson (2009)
11733:
11729:
11723:Eisenbud (1995)
11721:
11717:
11709:
11705:
11697:
11693:
11687:Bourbaki (1994)
11685:
11681:
11669:
11665:
11659:Bourbaki (1994)
11657:
11653:
11641:
11637:
11624:
11623:
11619:
11611:
11607:
11601:Bourbaki (1994)
11599:
11595:
11587:
11583:
11575:
11571:
11567:, Theorem 1.2.5
11563:
11559:
11551:
11547:
11539:
11535:
11527:
11523:
11517:Escofier (2012)
11515:
11511:
11503:
11499:
11491:
11487:
11483:, p. 120, Ch. 3
11479:
11475:
11467:
11463:
11455:
11451:
11447:, p. 120, Ch. 3
11443:
11439:
11427:
11423:
11415:
11411:
11403:
11399:
11393:Fraleigh (1976)
11391:
11387:
11379:
11375:
11371:
11366:
11365:
11356:
11352:
11338:
11332:
11326:
11322:
11316:
11307:
11306:
11300:
11294:
11292:
11288:
11275:
11269:
11267:
11263:
11254:
11249:or the maximal
11244:
11240:
11233:
11229:
11204:
11202:
11201:
11176:
11172:
11170:
11167:
11166:
11157:
11156:
11152:
11147:
11146:
11138:
11133:
11129:
11122:
11120:
11116:
11111:
11085:Michel Kervaire
11070:
11057:
11048:
11042:
11036:
11022:
11006:surreal numbers
10996:There are also
10978:
10972:
10969:
10961:
10953:
10950:
10948:Related notions
10942:elliptic curves
10935:
10924:
10900:
10892:
10884:
10881:
10880:
10870:
10864:
10854:
10843:
10831:
10822:
10816:
10798:
10786:
10776:
10768:
10762:
10748:
10745:
10737:
10731:
10709:
10688:
10677:
10669:
10668:
10660:
10659:
10658:that satisfies
10653:
10643:
10642:is a primitive
10640:
10632:
10628:
10616:
10573:
10565:
10557:
10554:
10546:
10540:
10522:
10503:
10497:
10479:
10473:
10462:
10447:
10422:
10412:
10406:
10402:
10393:
10383:
10377:
10367:
10357:
10345:
10331:
10322:
10286:
10272:
10270:
10268:
10265:
10264:
10240:
10203:
10202:
10195:
10189:
10168:
10133:
10108:
10104:to an equation
10099:
10092:
10084:
10073:
10067:
10049:
10036:
10030:
10024:
10018:
10011:
10000:
9943:
9939:
9931:
9928:
9927:
9921:
9915:
9902:
9889:
9886:
9881:
9868:
9862:
9854:
9848:
9833:
9827:
9787:
9782:
9763:
9759:
9748:
9733:
9728:
9722:
9719:
9718:
9633:
9629:
9624:
9618:
9614:
9599:
9595:
9577:
9572:
9566:
9563:
9562:
9556:Milnor K-theory
9553:
9543:
9528:
9517:
9516:central simple
9490:
9482:
9476:
9458:
9449:
9440:
9430:
9426:
9417:
9416:are the fields
9413:
9404:
9398:
9386:
9377:
9371:
9359:
9353:
9337:
9317:
9309:
9303:
9295:
9289:
9280:
9258:
9249:
9239:
9237:
9236:
9221:
9217:
9215:
9212:
9211:
9204:
9201:
9193:
9186:
9178:
9171:
9163:
9156:
9147:
9144:
9136:
9126:
9120:
9117:
9109:
9103:
9093:
9084:
9078:
9072:
9066:
9056:
9050:
9040:
9034:
9028:
9017:
9013:an integer, is
9008:
9001:
8997:
8993:
8983:
8977:
8967:
8961:
8950:
8935:
8932:
8923:
8914:
8913:
8899:
8893:
8887:
8881:
8871:
8868:
8845:
8836:
8827:
8821:
8811:
8796:
8790:
8787:
8779:
8775:
8765:
8755:
8749:
8746:
8735:
8734:
8720:
8711:
8699:
8680:
8666:
8663:
8657:
8636:
8631:
8629:
8626:
8625:
8616:
8598:
8586:
8568:
8554:
8545:
8539:
8523:
8513:
8507:
8501:
8495:
8489:
8483:
8467:
8458:
8429:
8412:
8406:
8382:
8372:
8336:
8332:
8327:
8323:
8319:
8315:
8294:
8289:
8288:
8287:
8283:
8257:
8253:
8248:
8244:
8240:
8236:
8230:
8225:
8224:
8223:
8219:
8211:
8208:
8207:
8195:
8187:
8181:
8178:
8170:
8164:
8158:
8146:
8138:
8126:
8118:
8115:
8107:
8091:
8083:
8076:
8068:
8065:
8057:
8045:
8033:
8030:
8018:
8008:
8005:
7997:
7989:
7981:
7975:
7967:
7954:
7946:
7938:
7927:
7919:
7916:
7907:
7906:
7898:
7894:
7886:
7875:
7863:
7855:
7846:
7844:
7834:
7825:
7815:
7811:
7803:
7796:
7793:-adic valuation
7781:
7771:
7748:
7746:
7739:
7713:
7693:
7691:
7678:
7672:
7662:
7656:
7650:
7648:
7643:
7637:
7628:
7622:
7609:
7583:
7570:continuous maps
7559:
7549:
7539:
7532:
7516:
7500:
7493:
7486:
7447:
7436:
7420:
7414:
7411:quadratic forms
7402:
7389:
7385:
7376:
7369:
7363:
7337:
7332:
7313:
7308:
7295:
7290:
7284:
7281:
7280:
7265:
7258:
7251:
7244:
7233:
7222:
7216:
7208:
7197:
7178:
7174:
7168:
7162:
7159:
7151:
7148:
7140:
7137:axiom of choice
7125:
7124:
7114:
7107:
7106:
7099:
7098:
7088:
7075:
7065:
7061:
7051:
7045:
7039:
7030:
7024:
7014:
7002:
6992:
6975:
6961:
6960:has a solution
6945:
6938:
6930:
6926:
6916:
6906:
6892:
6884:
6871:
6848:
6838:
6827:
6821:
6815:
6801:
6795:
6786:
6780:
6769:
6758:
6752:
6742:
6736:
6722:
6713:
6707:
6697:
6691:
6685:
6679:
6669:
6658:
6652:
6633:
6619:
6613:
6607:
6601:
6591:
6585:
6570:
6541:
6537:
6522:
6518:
6512:
6508:
6496:
6485:
6479:
6476:
6475:
6461:
6455:
6449:
6443:
6437:
6426:
6412:
6401:
6395:
6389:
6388:if and only if
6383:
6377:
6366:
6353:
6347:
6336:
6327:
6321:
6315:
6309:
6302:
6294:
6290:
6287:
6280:
6272:
6264:
6254:
6244:
6230:
6222:
6212:
6196:
6182:
6168:
6165:
6150:
6146:
6143:
6136:
6130:
6120:
6112:
6103:
6097:
6084:
6078:
6064:
6055:
6051:field extension
6043:
6033:
6030:
6024:
6013:
6007:
6001:
5996:
5992:
5986:
5980:
5974:
5969:
5965:
5959:
5941:
5935:
5929:
5923:
5913:
5907:
5889:
5883:
5872:
5866:
5856:
5850:
5844:
5838:
5832:
5826:
5820:
5814:
5811:
5787:
5776:
5771:
5769:
5768:
5748:
5744:
5743:
5739:
5734:
5720:
5718:
5715:
5714:
5699:
5698:
5683:
5682:
5676:
5670:
5654:
5653:
5641:
5640:
5636:
5622:
5616:
5600:
5599:
5590:
5580:
5574:
5559:
5550:
5537:
5531:
5525:
5515:
5509:
5499:
5493:
5481:
5471:
5463:
5457:
5447:
5436:
5435:
5426:
5416:
5407:
5404:
5393:
5386:
5375:
5369:
5340:
5336:
5328:
5310:
5306:
5300:
5296:
5290:
5279:
5273:
5270:
5269:
5252:
5246:
5244:polynomial ring
5237:
5222:
5192:
5175:
5173:
5160:
5147:
5145:
5142:
5141:
5128:
5118:
5108:
5101:
5095:
5089:
5083:
5073:
5062:
5052:
5041:
5032:
5030:integral domain
5026:
5018:
5010:
5004:
4998:
4992:
4986:
4980:
4974:
4963:
4957:
4953:
4939:
4929:
4923:
4913:
4907:
4900:
4895:
4858:Steinitz (1910)
4856:-adic numbers.
4851:
4840:
4832:
4815:
4809:
4802:Charles Hermite
4786:
4779:
4762:
4759:
4721:Évariste Galois
4712:
4697:
4689:
4677:
4667:
4640:
4634:
4616:
4609:
4599:
4592:
4585:
4575:
4568:
4561:
4555:
4540:
4527:
4524:
4516:
4510:
4499:
4495:
4491:
4485:
4482:
4476:
4470:
4460:
4457:
4451:
4444:
4438:
4432:
4422:
4416:
4410:
4407:
4399:
4384:
4374:
4373:
4364:splitting field
4353:
4344:
4330:
4323:
4317:
4307:
4301:
4295:
4287:
4281:
4275:
4262:
4249:
4232:
4221:
4208:
4194:
4188:
4177:
4174:
4168:
4161:
4151:
4145:
4139:
4118:
4109:
4103:
4091:
4081:
4077:
4073:
4067:
4063:
4059:
4048:
4044:
4041:
4035:
4032:
4026:
4016:
4010:
3999:
3996:
3988:
3982:
3976:
3970:
3957:
3951:
3950:(or the fields
3937:
3927:
3924:
3918:
3915:
3909:
3906:
3900:
3891:
3887:
3876:
3865:
3859:
3849:
3845:
3834:
3823:
3816:
3806:
3792:
3780:
3773:
3766:
3760:
3753:
3747:
3737:
3727:
3713:
3712:, that for all
3706:
3702:
3696:
3695:is a subset of
3690:
3689:. Equivalently
3684:
3678:
3677:is a subset of
3672:
3666:
3658:
3649:
3643:
3637:
3627:
3614:
3602:
3596:
3590:
3584:
3565:
3564:
3558:
3539:
3533:
3523:
3514:
3508:
3497:
3491:
3485:
3471:
3467:
3464:
3458:
3452:
3446:
3436:
3426:
3420:
3416:
3407:
3396:
3383:
3369:
3360:
3354:
3345:
3339:
3329:
3323:
3320:
3290:
3281:
3275:
3268:
3232:
3229:
3228:
3222:
3216:
3207:
3179:
3176:
3175:
3135:
3132:
3131:
3121:
3116:Similarly, the
3108:
3100:
3086:
3083:
3071:
3060:
3031:
3019:
3014:
3009:
3000:
2994:
2988:integral domain
2957:
2950:
2946:
2940:
2934:
2927:
2920:
2909:
2902:
2899:
2888:
2878:
2872:
2866:
2863:
2854:
2851:
2845:
2834:
2828:
2820:
2817:
2811:
2768:
2767:
2766:, which equals
2741:
2734:
2728:
2722:
2716:
2710:
2704:
2698:
2683:
2676:
2669:
2662:
2655:
2646:
2639:
2632:
2625:
2618:
2609:
2602:
2594:
2593:
2585:
2584:
2577:
2568:
2561:
2553:
2552:
2544:
2543:
2536:
2527:
2520:
2513:
2506:
2488:
2481:
2474:
2467:
2460:
2451:
2444:
2437:
2430:
2423:
2414:
2407:
2399:
2398:
2390:
2389:
2382:
2373:
2366:
2358:
2357:
2349:
2348:
2341:
2332:
2325:
2318:
2311:
2299:Multiplication
2289:
2283:
2256:
2251:
2249:
2246:
2245:
2236:
2230:
2212:
2204:
2201:
2200:
2194:
2188:
2178:
2174:(center at the
2169:
2159:
2153:
2147:
2137:
2118:
2105:
2098:
2088:
2077:
2004:
1995:
1988:
1978:
1964:
1953:
1944:
1942:complex numbers
1935:
1929:
1921:Main articles:
1911:
1892:
1891:
1878:
1865:
1852:
1839:
1837:
1835:
1829:
1828:
1818:
1810:
1808:
1795:
1787:
1785:
1769:
1758:
1756:
1740:
1729:
1727:
1711:
1685:
1683:
1681:
1679:
1673:
1672:
1662:
1645:
1643:
1630:
1612:
1604:
1602:
1589:
1581:
1579:
1578:
1574:
1561:
1559:
1557:
1551:
1550:
1535:
1522:
1509:
1496:
1495:
1491:
1478:
1476:
1474:
1468:
1467:
1452:
1439:
1438:
1434:
1421:
1416:
1414:
1411:
1410:
1377:
1369:
1367:
1354:
1341:
1339:
1336:
1335:
1322:
1315:
1304:
1297:
1287:
1281:
1271:
1265:
1263:Rational number
1259:
1254:
1238:
1234:
1230:
1226:
1214:
1210:
1194:
1181:
1170:
1166:
1127:
1110:
1104:
1093:
1087:
1081:
1075:
1068:
1051:
1045:
1034:
1028:
1022:
1016:
1000:
990:
984:
980:
976:
948:
930:
898:
872:
861:
853:
849:
845:
829:
823:
817:
811:
805:
799:
789:
783:
777:
771:
765:
761:
757:
743:
739:
727:
717:
711:
687:
666:
662:, by defining:
653:
643:
635:
629:
622:
615:
594:
584:
577:
569:Function fields
468:complex numbers
462:, the field of
401:
372:
371:
370:
341:Non-associative
323:
312:
311:
301:
281:
270:
269:
258:Map of lattices
254:
250:Boolean algebra
245:Heyting algebra
219:
208:
207:
201:
182:Integral domain
146:
135:
134:
128:
82:
39:
28:
23:
22:
18:Field (algebra)
15:
12:
11:
5:
13728:
13718:
13717:
13712:
13707:
13692:
13691:External links
13689:
13687:
13686:
13653:(4): 521–549,
13635:
13622:
13598:
13585:
13572:
13563:
13549:
13538:
13526:
13473:
13460:
13443:
13422:
13398:
13385:
13369:
13356:
13340:
13327:
13304:
13299:
13286:
13281:
13260:Scholze, Peter
13256:
13243:
13219:
13206:
13185:
13150:
13137:
13116:
13096:
13083:
13074:10.1.1.36.8448
13056:
13051:
13038:
13025:
13011:Lidl, Rudolf;
13008:
13003:
12979:
12966:
12953:
12917:
12904:
12890:, Birkhäuser,
12880:
12832:
12827:
12811:
12776:
12758:
12753:
12737:
12722:
12701:Fricke, Robert
12697:
12692:
12684:Addison-Wesley
12675:
12670:
12657:
12644:
12612:
12594:
12581:
12565:
12562:Academic Press
12548:
12543:
12528:
12515:
12491:
12486:
12470:
12457:
12433:
12420:
12407:
12398:) and Book 6 (
12377:Blyth, T. S.;
12374:
12369:
12356:
12339:(4): 383–385,
12328:
12301:(2): 145–205,
12285:
12268:(2): 239–271,
12254:
12211:Schreier, Otto
12203:
12197:
12181:Artin, Michael
12177:
12172:
12159:
12154:
12140:
12138:
12135:
12133:
12132:
12121:
12110:
12099:
12088:
12076:
12065:
12053:
12041:
12030:
12019:
12008:
11996:
11984:
11972:
11956:
11944:
11932:
11920:
11908:
11905:Scholze (2014)
11897:
11886:
11874:
11862:
11860:, Theorem 1.23
11858:Prestel (1984)
11850:
11846:Prestel (1984)
11838:
11826:
11814:
11812:, p. 186, §7.1
11802:
11787:
11775:
11763:
11751:
11739:
11727:
11715:
11703:
11691:
11679:
11663:
11651:
11647:Kleiner (2007)
11635:
11617:
11605:
11593:
11589:Kiernan (1971)
11581:
11577:Kleiner (2007)
11569:
11557:
11545:
11541:Adamson (2007)
11533:
11529:Adamson (2007)
11521:
11509:
11505:Adamson (2007)
11497:
11493:Wallace (1998)
11485:
11473:
11471:, Example 1.62
11461:
11459:, Chapter 13.4
11449:
11437:
11421:
11409:
11397:
11385:
11372:
11370:
11367:
11364:
11363:
11350:
11320:
11312:
11302:
11286:
11261:
11238:
11216:
11210:
11207:
11200:
11197:
11194:
11191:
11188:
11185:
11182:
11179:
11175:
11127:
11113:
11112:
11110:
11107:
11103:division rings
11021:
11020:Division rings
11018:
10998:proper classes
10965:
10949:
10946:
10912:
10907:
10904:
10899:
10895:
10891:
10888:
10851:
10850:
10839:
10807:describes the
10772:
10741:
10724:
10723:
10636:
10624:
10595:Pierre Deligne
10561:
10550:
10521:
10518:
10398:
10391:
10316:
10315:
10304:
10298:
10295:
10292:
10289:
10284:
10281:
10278:
10275:
10237:
10236:
10185:on a suitable
10167:
10164:
10152:
10151:
10127:elliptic curve
10119:
10118:
10088:
10081:
10080:
10010:
10007:
9987:The theory of
9970:linear algebra
9957:
9954:
9949:
9946:
9942:
9938:
9935:
9912:
9911:
9885:
9882:
9880:
9877:
9866:
9852:
9831:
9813:
9812:
9801:
9798:
9793:
9790:
9785:
9781:
9777:
9774:
9771:
9766:
9762:
9758:
9755:
9751:
9747:
9744:
9741:
9736:
9731:
9727:
9704:
9703:
9692:
9688:
9684:
9681:
9678:
9675:
9672:
9669:
9666:
9663:
9660:
9657:
9654:
9651:
9648:
9645:
9642:
9639:
9636:
9632:
9627:
9621:
9617:
9613:
9610:
9607:
9602:
9598:
9594:
9591:
9588:
9585:
9580:
9575:
9571:
9558:is defined as
9552:
9549:
9548:
9547:
9541:
9486:
9473:
9472:
9454:
9445:
9422:
9409:
9382:
9352:
9349:
9326:
9325:
9313:
9305:
9299:
9291:
9277:
9276:
9265:
9261:
9257:
9252:
9246:
9242:
9235:
9230:
9227:
9224:
9220:
9197:
9182:
9175:
9174:
9167:
9158:
9140:
9113:
9047:
9046:
9039:has a zero in
8956:, a branch of
8949:
8946:
8928:
8919:
8867:
8864:
8856:
8855:
8841:
8804:
8803:
8783:
8770:
8763:
8744:
8729:
8709:
8659:
8639:
8613:abelian groups
8480:
8479:
8408:Main article:
8405:
8402:
8371:
8368:
8367:
8366:
8355:
8351:
8346:
8339:
8335:
8330:
8326:
8322:
8318:
8314:
8311:
8308:
8305:
8302:
8297:
8292:
8286:
8282:
8279:
8276:
8272:
8267:
8260:
8256:
8251:
8247:
8243:
8239:
8233:
8228:
8222:
8218:
8215:
8191:
8174:
8155:
8149:
8142:
8122:
8111:
8087:
8078:is done using
8072:
8061:
8041:
8026:
8015:
8014:
8001:
7985:
7978:
7971:
7953:
7950:
7942:
7923:
7912:
7903:-adic analysis
7890:
7881:
7880:
7873:
7861:
7852:
7831:
7830:
7823:
7807:
7801:
7786:
7778:
7777:
7769:
7764:
7761:absolute value
7744:
7736:
7735:
7734:zero sequence
7732:
7729:
7726:
7689:absolute value
7602:
7601:
7531:
7528:
7483:bounded subset
7444:infinitesimals
7440:
7439:
7381:
7374:
7367:
7360:
7359:
7348:
7345:
7340:
7335:
7331:
7327:
7324:
7321:
7316:
7311:
7307:
7303:
7298:
7293:
7289:
7218:Main article:
7215:
7214:Ordered fields
7212:
7207:
7204:
7194:Puiseux series
7190:Laurent series
7155:
7144:
7038:By the above,
6999:
6998:
6958:
6957:
6943:
6934:
6924:
6914:
6901:
6888:
6870:
6867:
6704:transcendental
6690:whose zero is
6651:
6648:
6567:
6566:
6555:
6552:
6549:
6544:
6540:
6530:
6525:
6521:
6515:
6511:
6505:
6502:
6499:
6494:
6491:
6488:
6484:
6471:have the form
6344:
6343:
6307:imaginary unit
6298:
6285:
6276:
6269:
6268:
6262:
6252:
6239:
6226:
6164:
6161:
6145:are of degree
6141:
6134:
6117:
6116:
6075:
6074:
6023:
6020:
5973:of some field
5819:, and a field
5810:
5807:
5806:
5805:
5794:
5790:
5779:
5774:
5763:
5759:
5756:
5751:
5747:
5742:
5737:
5733:
5730:
5727:
5723:
5633:imaginary unit
5613:
5612:
5571:
5570:
5508:(over a field
5423:surjective map
5421:by means of a
5403:
5402:Residue fields
5400:
5366:
5365:
5354:
5351:
5348:
5343:
5339:
5335:
5331:
5327:
5324:
5321:
5313:
5309:
5303:
5299:
5293:
5288:
5285:
5282:
5278:
5264:Laurent series
5216:
5215:
5204:
5198:
5195:
5190:
5187:
5184:
5181:
5178:
5172:
5167:
5164:
5159:
5154:
5151:
5025:
5022:
5014:
4922:of an integer
4899:
4896:
4894:
4891:
4836:
4757:
4674:
4673:
4631:François Viète
4621:being a third
4613:
4612:
4607:
4597:
4590:
4573:
4566:
4559:
4539:
4536:
4520:
4480:
4455:
4403:
4396:
4395:
4291:
4172:
4136:
4135:
4129:= {0, 1, ...,
4039:
4030:
4012:Main article:
4009:
4006:
3992:
3922:
3913:
3902:
3896:
3885:
3874:
3863:
3857:
3843:
3832:
3821:
3814:
3701:that contains
3657:
3654:
3634:
3633:
3547:
3546:
3462:
3414:characteristic
3404:
3403:
3390:
3389:
3350:by a positive
3319:
3318:Characteristic
3316:
3248:
3245:
3242:
3239:
3236:
3195:
3192:
3189:
3186:
3183:
3163:
3160:
3157:
3154:
3151:
3148:
3145:
3142:
3139:
3097:additive group
3082:
3079:
3078:
3077:
3058:
3029:
3017:
3012:
2898:
2895:
2883:are arbitrary
2862:
2859:
2849:
2827:consisting of
2815:
2800:
2799:
2693:
2692:
2689:
2688:
2681:
2674:
2667:
2660:
2652:
2651:
2644:
2637:
2630:
2623:
2615:
2614:
2607:
2600:
2591:
2582:
2574:
2573:
2566:
2559:
2550:
2541:
2533:
2532:
2525:
2518:
2511:
2504:
2497:
2494:
2493:
2486:
2479:
2472:
2465:
2457:
2456:
2449:
2442:
2435:
2428:
2420:
2419:
2412:
2405:
2396:
2387:
2379:
2378:
2371:
2364:
2355:
2346:
2338:
2337:
2330:
2323:
2316:
2309:
2301:
2300:
2297:
2285:Main article:
2282:
2279:
2259:
2255:
2216:
2211:
2208:
2114:Main article:
2076:
2073:
2060:
2059:
1985:imaginary unit
1975:
1974:
1927:Complex number
1910:
1907:
1906:
1905:
1890:
1885:
1882:
1877:
1872:
1869:
1864:
1859:
1856:
1851:
1846:
1843:
1838:
1834:
1831:
1830:
1824:
1821:
1816:
1813:
1807:
1801:
1798:
1793:
1790:
1784:
1778:
1775:
1772:
1767:
1764:
1761:
1755:
1749:
1746:
1743:
1738:
1735:
1732:
1726:
1720:
1717:
1714:
1709:
1706:
1703:
1700:
1697:
1694:
1691:
1688:
1682:
1678:
1675:
1674:
1668:
1665:
1660:
1657:
1654:
1651:
1648:
1642:
1637:
1634:
1629:
1625:
1618:
1615:
1610:
1607:
1601:
1595:
1592:
1587:
1584:
1577:
1573:
1568:
1565:
1560:
1556:
1553:
1552:
1548:
1542:
1539:
1534:
1529:
1526:
1521:
1516:
1513:
1508:
1503:
1500:
1494:
1490:
1485:
1482:
1477:
1473:
1470:
1469:
1465:
1459:
1456:
1451:
1446:
1443:
1437:
1433:
1428:
1425:
1420:
1418:
1404:
1403:
1392:
1389:
1383:
1380:
1375:
1372:
1366:
1361:
1358:
1353:
1348:
1345:
1261:Main article:
1258:
1255:
1253:
1250:
1193:
1190:
1159:
1158:
1124:Distributivity
1121:
1062:
1010:
966:
924:
856:are arbitrary
736:multiplication
710:
707:
706:
705:
685:
576:
573:
557:linear algebra
429:multiplication
403:
402:
400:
399:
392:
385:
377:
374:
373:
369:
368:
363:
358:
353:
348:
343:
338:
332:
331:
330:
324:
318:
317:
314:
313:
310:
309:
306:Linear algebra
300:
299:
294:
289:
283:
282:
276:
275:
272:
271:
268:
267:
264:Lattice theory
260:
253:
252:
247:
242:
237:
232:
227:
221:
220:
214:
213:
210:
209:
200:
199:
194:
189:
184:
179:
174:
169:
164:
159:
154:
148:
147:
141:
140:
137:
136:
127:
126:
121:
116:
110:
109:
108:
103:
98:
89:
83:
77:
76:
73:
72:
26:
9:
6:
4:
3:
2:
13727:
13716:
13713:
13711:
13708:
13706:
13703:
13702:
13700:
13684:
13680:
13676:
13672:
13668:
13664:
13660:
13656:
13652:
13649:(in German),
13648:
13644:
13640:
13636:
13633:
13629:
13625:
13623:0-387-94762-0
13619:
13615:
13611:
13607:
13603:
13599:
13596:
13592:
13588:
13586:0-444-87429-1
13582:
13578:
13573:
13569:
13564:
13560:
13559:
13554:
13550:
13546:
13545:
13539:
13535:
13531:
13530:Tits, Jacques
13527:
13524:
13520:
13516:
13512:
13508:
13504:
13500:
13496:
13492:
13488:
13487:
13482:
13478:
13474:
13471:
13467:
13463:
13461:0-521-33718-6
13457:
13452:
13451:
13444:
13441:
13437:
13433:
13429:
13425:
13419:
13415:
13411:
13407:
13403:
13399:
13396:
13392:
13388:
13386:0-86720-210-6
13382:
13378:
13374:
13370:
13367:
13363:
13359:
13357:0-387-90424-7
13353:
13349:
13345:
13341:
13338:
13334:
13330:
13328:9780387900407
13324:
13319:
13318:
13315:
13309:
13305:
13302:
13296:
13292:
13287:
13284:
13278:
13274:
13273:
13265:
13261:
13257:
13254:
13250:
13246:
13244:0-387-98525-5
13240:
13236:
13232:
13228:
13224:
13220:
13217:
13213:
13209:
13207:3-540-13885-4
13203:
13199:
13195:
13191:
13186:
13183:
13179:
13174:
13169:
13165:
13161:
13160:
13155:
13151:
13148:
13144:
13140:
13138:0-387-96640-4
13134:
13130:
13126:
13122:
13117:
13114:
13110:
13106:
13102:
13097:
13094:
13090:
13086:
13080:
13075:
13070:
13065:
13064:
13057:
13054:
13048:
13044:
13039:
13036:
13032:
13028:
13022:
13018:
13017:Finite fields
13014:
13009:
13006:
13004:0-387-95385-X
13000:
12996:
12992:
12988:
12984:
12980:
12977:
12973:
12969:
12967:0-8218-0943-1
12963:
12959:
12954:
12951:
12947:
12943:
12939:
12935:
12931:
12927:
12923:
12918:
12915:
12911:
12907:
12901:
12897:
12893:
12889:
12885:
12881:
12878:
12874:
12870:
12866:
12862:
12858:
12854:
12850:
12846:
12842:
12841:Invent. Math.
12838:
12833:
12830:
12824:
12820:
12819:Basic algebra
12816:
12812:
12809:
12805:
12801:
12797:
12793:
12790:(in German),
12789:
12785:
12781:
12777:
12773:
12769:
12768:
12763:
12759:
12756:
12750:
12746:
12742:
12738:
12734:
12733:-adic numbers
12730:
12727:
12723:
12720:
12716:
12712:
12711:
12706:
12702:
12698:
12695:
12693:0-201-01984-1
12689:
12685:
12681:
12676:
12673:
12667:
12663:
12662:Galois Theory
12658:
12655:
12651:
12647:
12645:0-387-94268-8
12641:
12637:
12633:
12629:
12625:
12621:
12617:
12613:
12609:
12608:
12603:
12599:
12595:
12592:
12588:
12584:
12582:3-7643-7002-5
12578:
12574:
12570:
12566:
12563:
12559:
12558:
12553:
12549:
12546:
12540:
12536:
12535:
12529:
12526:
12522:
12518:
12516:0-521-30484-9
12512:
12508:
12504:
12500:
12496:
12492:
12489:
12487:0-387-19375-8
12483:
12479:
12475:
12471:
12468:
12464:
12460:
12458:3-540-19376-6
12454:
12450:
12446:
12442:
12438:
12434:
12431:
12427:
12423:
12421:0-521-80309-8
12417:
12413:
12408:
12405:
12404:0-521-27291-2
12401:
12397:
12396:0-521-27288-2
12393:
12388:
12384:
12380:
12375:
12372:
12370:1-57766-443-4
12366:
12362:
12357:
12354:
12350:
12346:
12342:
12338:
12334:
12329:
12326:
12322:
12318:
12314:
12309:
12304:
12300:
12296:
12295:
12290:
12289:Baez, John C.
12286:
12283:
12279:
12275:
12271:
12267:
12263:
12262:Ann. of Math.
12259:
12255:
12252:
12248:
12244:
12240:
12236:
12232:
12228:
12224:
12220:
12217:(in German),
12216:
12212:
12208:
12204:
12200:
12194:
12190:
12189:Prentice Hall
12186:
12182:
12178:
12175:
12169:
12165:
12160:
12157:
12151:
12147:
12142:
12141:
12130:
12125:
12119:
12118:Conway (1976)
12114:
12108:
12103:
12097:
12092:
12085:
12080:
12074:
12069:
12062:
12057:
12050:
12045:
12039:
12034:
12028:
12023:
12017:
12012:
12005:
12000:
11993:
11988:
11981:
11980:Gouvêa (2012)
11976:
11969:
11965:
11960:
11953:
11948:
11941:
11936:
11929:
11924:
11917:
11912:
11906:
11901:
11895:
11890:
11883:
11882:Gouvêa (1997)
11878:
11871:
11870:Warner (1989)
11866:
11859:
11854:
11847:
11842:
11835:
11834:Lorenz (2008)
11830:
11823:
11818:
11811:
11806:
11800:
11796:
11791:
11784:
11779:
11772:
11767:
11760:
11755:
11748:
11743:
11736:
11731:
11724:
11719:
11712:
11707:
11700:
11695:
11688:
11683:
11676:
11672:
11667:
11660:
11655:
11648:
11644:
11639:
11631:
11629:
11621:
11614:
11609:
11602:
11597:
11590:
11585:
11578:
11573:
11566:
11561:
11554:
11549:
11542:
11537:
11530:
11525:
11518:
11513:
11506:
11501:
11494:
11489:
11482:
11477:
11470:
11465:
11458:
11453:
11446:
11441:
11434:
11433:Heyting field
11430:
11425:
11418:
11413:
11406:
11401:
11394:
11389:
11382:
11377:
11373:
11360:
11354:
11345:
11341:
11335:
11330:
11324:
11315:
11310:
11305:
11297:
11290:
11283:
11282:Cassels (1986
11278:
11272:
11265:
11257:
11252:
11248:
11242:
11214:
11208:
11205:
11198:
11195:
11192:
11186:
11180:
11177:
11173:
11161:
11142:
11137:
11131:
11118:
11114:
11106:
11104:
11100:
11096:
11094:
11090:
11086:
11082:
11078:
11073:
11069:
11065:
11060:
11056:
11051:
11045:
11039:
11034:
11030:
11029:
11028:division ring
11017:
11015:
11011:
11007:
11003:
10999:
10994:
10992:
10988:
10984:
10975:
10968:
10964:
10959:
10945:
10943:
10938:
10933:
10927:
10905:
10902:
10889:
10886:
10878:
10873:
10867:
10862:
10857:
10847:
10842:
10838:
10834:
10830:
10829:
10828:
10825:
10821:extension of
10819:
10814:
10810:
10806:
10801:
10794:
10790:
10784:
10779:
10775:
10771:
10765:
10760:
10756:
10751:
10744:
10740:
10734:
10729:
10720:
10716:
10712:
10708:
10707:
10706:
10704:
10700:
10696:
10695:regular prime
10691:
10685:
10681:
10672:
10663:
10656:
10651:
10650:root of unity
10646:
10639:
10635:
10627:
10623:
10619:
10614:
10607:
10602:
10598:
10596:
10592:
10588:
10584:
10579:
10569:
10564:
10560:
10553:
10549:
10543:
10538:
10537:number fields
10534:
10530:
10526:
10525:Global fields
10517:
10515:
10511:
10506:
10500:
10495:
10491:
10487:
10482:
10476:
10469:
10465:
10460:
10454:
10450:
10445:
10441:
10437:
10433:
10428:
10425:
10420:
10415:
10409:
10401:
10397:
10390:
10386:
10380:
10376:over a field
10375:
10372:-dimensional
10370:
10365:
10360:
10356:
10351:
10348:
10343:
10339:
10334:
10330:
10325:
10319:
10302:
10293:
10287:
10279:
10273:
10263:
10262:
10261:
10259:
10255:
10252:For having a
10250:
10248:
10243:
10231:
10227:
10223:
10219:
10215:
10211:
10207:
10201:
10200:
10199:
10194:into a field
10192:
10188:
10184:
10177:
10172:
10163:
10161:
10160:combinatorics
10157:
10156:coding theory
10148:
10144:
10140:
10136:
10132:
10131:
10130:
10128:
10124:
10115:
10111:
10107:
10106:
10105:
10102:
10097:
10091:
10087:
10076:
10070:
10064:
10060:
10056:
10052:
10048:
10047:
10046:
10039:
10033:
10027:
10021:
10015:
10006:
10003:
9998:
9994:
9990:
9985:
9983:
9979:
9975:
9971:
9955:
9952:
9947:
9944:
9940:
9936:
9933:
9924:
9918:
9909:
9905:
9901:
9900:
9899:
9898:
9892:
9876:
9872:
9865:
9858:
9851:
9846:
9841:
9837:
9830:
9825:
9821:
9817:
9799:
9791:
9788:
9783:
9779:
9775:
9772:
9764:
9760:
9756:
9753:
9749:
9742:
9734:
9729:
9725:
9717:
9716:
9715:
9713:
9709:
9690:
9686:
9679:
9676:
9673:
9667:
9664:
9661:
9658:
9655:
9649:
9646:
9643:
9637:
9634:
9630:
9625:
9619:
9615:
9611:
9608:
9605:
9600:
9596:
9592:
9586:
9578:
9573:
9569:
9561:
9560:
9559:
9557:
9540:
9536:
9532:
9527:
9526:
9525:
9523:
9520:
9513:
9509:
9505:
9501:
9497:
9493:
9489:
9485:
9479:
9469:
9466:
9462:
9457:
9453:
9448:
9444:
9439:
9438:
9437:
9433:
9425:
9421:
9412:
9408:
9401:
9396:
9392:
9385:
9381:
9374:
9369:
9363:
9358:
9348:
9345:
9341:
9335:
9331:
9321:
9316:
9312:
9308:
9302:
9298:
9294:
9288:
9287:
9286:
9283:
9263:
9255:
9250:
9233:
9222:
9218:
9210:
9209:
9208:
9200:
9196:
9191:
9190:Łoś's theorem
9185:
9181:
9170:
9166:
9161:
9155:
9154:
9153:
9150:
9143:
9139:
9134:
9129:
9123:
9116:
9112:
9106:
9101:
9096:
9090:
9087:
9081:
9075:
9069:
9064:
9059:
9053:
9043:
9037:
9031:
9024:
9020:
9016:
9015:
9014:
9011:
9004:
8991:
8986:
8980:
8975:
8970:
8964:
8960:, two fields
8959:
8955:
8945:
8943:
8938:
8931:
8927:
8922:
8917:
8911:
8907:
8902:
8896:
8890:
8884:
8879:
8874:
8863:
8861:
8852:
8848:
8844:
8839:
8835:
8834:
8833:
8830:
8824:
8818:
8814:
8809:
8799:
8793:
8786:
8782:
8773:
8769:
8762:
8758:
8752:
8743:
8738:
8732:
8728:
8723:
8718:
8714:
8710:
8706:
8702:
8695:
8691:
8687:
8683:
8679:
8678:
8677:
8675:
8669:
8662:
8656:
8637:
8623:
8619:
8614:
8610:
8605:
8601:
8594:
8590:
8584:
8580:
8576:
8571:
8565:
8561:
8557:
8553:
8548:
8542:
8537:
8531:
8527:
8522:
8517:
8510:
8504:
8498:
8492:
8486:
8475:
8470:
8465:
8461:
8457:
8456:
8455:
8453:
8449:
8445:
8441:
8436:
8432:
8428:
8425:
8421:
8417:
8411:
8410:Galois theory
8404:Galois theory
8401:
8399:
8395:
8389:
8385:
8380:
8376:
8353:
8349:
8344:
8333:
8328:
8324:
8320:
8316:
8306:
8295:
8284:
8280:
8277:
8274:
8270:
8265:
8254:
8249:
8245:
8241:
8237:
8231:
8220:
8216:
8213:
8205:
8199:
8194:
8190:
8184:
8177:
8173:
8167:
8161:
8156:
8153:
8150:
8145:
8141:
8136:
8130:
8125:
8121:
8114:
8110:
8105:
8101:
8100:
8099:
8095:
8090:
8086:
8081:
8075:
8071:
8064:
8060:
8055:
8049:
8044:
8040:
8036:
8029:
8025:
8021:
8011:
8004:
8000:
7993:
7988:
7984:
7979:
7974:
7970:
7965:
7964:
7963:
7961:
7960:
7949:
7945:
7941:
7936:
7935:-adic numbers
7934:
7926:
7922:
7915:
7910:
7904:
7901:
7893:
7889:
7878:
7874:
7870:
7866:
7862:
7858:
7853:
7849:
7841:
7837:
7833:
7832:
7828:
7824:
7821:
7820:-adic numbers
7818:
7810:
7806:
7802:
7799:
7794:
7792:
7787:
7784:
7780:
7779:
7775:
7770:
7768:
7765:
7762:
7755:
7751:
7745:
7742:
7738:
7737:
7733:
7730:
7727:
7724:
7723:
7720:
7716:
7711:
7704:
7700:
7696:
7690:
7687:given by the
7685:
7681:
7675:
7669:
7665:
7659:
7646:
7640:
7636:
7631:
7625:
7620:
7615:
7612:
7607:
7598:
7594:
7590:
7586:
7582:
7581:
7580:
7579:
7575:
7571:
7566:
7562:
7556:
7552:
7547:
7542:
7537:
7527:
7525:
7519:
7515:
7510:
7508:
7503:
7497:
7489:
7484:
7480:
7479:
7474:
7470:
7466:
7457:
7453:
7450:
7445:
7437:1 + 1 + ⋯ + 1
7435:
7434:
7433:
7431:
7426:
7423:
7417:
7412:
7406:
7401:
7397:
7392:
7384:
7380:
7373:
7366:
7346:
7343:
7338:
7333:
7329:
7325:
7322:
7319:
7314:
7309:
7305:
7301:
7296:
7291:
7287:
7279:
7278:
7277:
7275:
7271:
7261:
7254:
7247:
7240:
7236:
7231:
7230:ordered field
7228:is called an
7227:
7221:
7220:Ordered field
7211:
7203:
7200:
7195:
7191:
7185:
7181:
7171:
7165:
7158:
7154:
7147:
7143:
7138:
7134:
7128:
7122:
7117:
7110:
7102:
7096:
7091:
7085:
7083:
7078:
7073:
7068:
7059:
7054:
7048:
7042:
7036:
7033:
7027:
7022:
7017:
7012:
7011:
7007:is called an
7005:
6995:
6991:
6990:
6989:
6987:
6983:
6978:
6973:
6968:
6964:
6953:
6949:
6942:
6937:
6933:
6923:
6919:
6913:
6909:
6904:
6900:
6896:
6891:
6887:
6883:
6882:
6881:
6880:
6876:
6866:
6864:
6859:
6855:
6851:
6845:
6841:
6834:
6830:
6824:
6818:
6813:
6809:
6804:
6798:
6792:
6789:
6783:
6776:
6772:
6765:
6761:
6755:
6750:
6745:
6739:
6733:
6729:
6725:
6719:
6716:
6710:
6705:
6700:
6694:
6688:
6682:
6677:
6676:indeterminate
6672:
6665:
6661:
6657:
6647:
6644:
6640:
6636:
6630:
6626:
6622:
6616:
6610:
6604:
6598:
6594:
6588:
6583:
6577:
6573:
6553:
6550:
6547:
6542:
6538:
6528:
6523:
6519:
6513:
6509:
6503:
6500:
6497:
6492:
6489:
6486:
6482:
6474:
6473:
6472:
6468:
6464:
6458:
6452:
6446:
6440:
6433:
6429:
6423:
6419:
6415:
6408:
6404:
6398:
6392:
6386:
6380:
6373:
6369:
6365:The subfield
6363:
6361:
6358:is called an
6356:
6350:
6339:
6335:
6334:
6333:
6330:
6324:
6318:
6312:
6308:
6301:
6297:
6284:
6279:
6275:
6261:
6257:
6251:
6247:
6242:
6238:
6234:
6229:
6225:
6221:
6220:
6219:
6218:
6210:
6206:
6202:
6194:
6189:
6185:
6181:. An element
6180:
6175:
6171:
6160:
6157:
6153:
6140:
6133:
6127:
6123:
6111:
6110:
6109:
6106:
6100:
6095:
6090:
6087:
6081:
6071:
6067:
6063:
6062:
6061:
6060:, denoted by
6058:
6053:
6052:
6046:
6040:
6036:
6029:
6019:
6016:
6010:
6004:
5995:
5989:
5983:
5977:
5968:
5962:
5957:
5952:
5948:
5944:
5940:, denoted by
5938:
5932:
5926:
5920:
5916:
5910:
5905:
5903:
5896:
5892:
5886:
5879:
5875:
5869:
5865:generated by
5864:
5859:
5853:
5847:
5841:
5835:
5829:
5823:
5817:
5792:
5777:
5761:
5757:
5754:
5749:
5745:
5740:
5735:
5728:
5713:
5712:
5711:
5707:
5702:
5695:
5691:
5686:
5679:
5673:
5666:
5662:
5657:
5649:
5644:
5634:
5630:
5625:
5619:
5615:For example,
5608:
5603:
5598:
5597:
5596:
5593:
5588:
5587:residue class
5583:
5577:
5567:
5562:
5557:
5553:
5549:
5548:
5547:
5545:
5540:
5534:
5528:
5523:
5518:
5512:
5506:
5502:
5496:
5492:
5487:
5484:
5479:
5478:residue field
5474:
5470:
5466:
5460:
5455:
5454:maximal ideal
5450:
5443:
5439:
5434:
5429:
5425:onto a field
5424:
5419:
5414:
5410:
5399:
5396:
5389:
5384:
5378:
5372:
5368:over a field
5349:
5346:
5341:
5337:
5333:
5325:
5322:
5311:
5307:
5301:
5297:
5286:
5283:
5280:
5276:
5268:
5267:
5266:
5265:
5259:
5255:
5249:
5245:
5240:
5235:
5229:
5225:
5219:
5202:
5196:
5193:
5188:
5185:
5182:
5179:
5176:
5170:
5165:
5162:
5157:
5152:
5149:
5140:
5139:
5138:
5135:
5131:
5125:
5121:
5115:
5111:
5104:
5098:
5092:
5086:
5080:
5076:
5069:
5065:
5060:
5055:
5048:
5044:
5040:
5035:
5031:
5021:
5017:
5013:
5007:
5001:
4995:
4989:
4983:
4977:
4971:
4969:
4960:
4951:
4947:
4942:
4936:
4932:
4926:
4921:
4916:
4910:
4905:
4890:
4888:
4884:
4880:
4876:
4872:
4868:
4864:
4863:Galois theory
4859:
4854:
4849:
4848:Hensel (1904)
4845:
4839:
4835:
4830:
4826:
4821:
4818:
4812:
4807:
4803:
4799:
4793:
4789:
4782:
4776:
4772:
4768:
4756:
4751:
4749:
4745:
4741:
4737:
4732:
4730:
4726:
4725:Galois theory
4722:
4718:
4710:
4706:
4705:Paolo Ruffini
4700:
4695:
4692:
4685:
4680:
4670:
4666:
4665:
4664:
4662:
4661:
4656:
4652:
4648:
4643:
4637:
4632:
4628:
4624:
4623:root of unity
4619:
4606:
4603:
4596:
4589:
4584:
4583:
4582:
4580:
4572:
4565:
4558:
4553:
4549:
4545:
4535:
4531:
4523:
4519:
4513:
4508:
4502:
4488:
4479:
4473:
4467:
4463:
4454:
4447:
4441:
4435:
4430:
4425:
4419:
4413:
4406:
4402:
4392:
4387:
4382:
4377:
4372:
4371:
4370:
4369:
4365:
4360:
4356:
4352:A field with
4350:
4347:
4342:
4338:
4333:
4326:
4322:is prime and
4320:
4314:
4310:
4304:
4298:
4294:
4290:
4284:
4278:
4272:
4269:
4265:
4259:
4256:
4252:
4247:
4242:
4239:
4235:
4228:
4224:
4218:
4215:
4211:
4205:
4201:
4197:
4191:
4186:
4180:
4171:
4164:
4159:
4154:
4148:
4142:
4132:
4128:
4125:
4121:
4117:
4116:
4115:
4112:
4106:
4101:
4088:
4084:
4070:
4056:
4052:
4038:
4029:
4024:
4023:Galois fields
4021:(also called
4020:
4019:Finite fields
4015:
4008:Finite fields
4005:
4002:
3995:
3991:
3985:
3979:
3973:
3968:
3963:
3960:
3954:
3949:
3945:
3940:
3935:
3930:
3921:
3912:
3905:
3899:
3894:
3884:
3880:
3873:
3869:
3862:
3856:
3852:
3842:
3838:
3831:
3827:
3820:
3813:
3809:
3803:
3799:
3795:
3790:
3786:
3783:
3777:
3770:
3763:
3756:
3750:
3744:
3740:
3734:
3730:
3724:
3720:
3716:
3710:
3699:
3693:
3687:
3681:
3675:
3669:
3665:
3664:
3653:
3646:
3640:
3630:
3625:
3621:
3617:
3613:
3612:
3611:
3610:
3609:Frobenius map
3605:
3599:
3593:
3587:
3581:
3577:
3573:
3568:
3561:
3556:
3552:
3542:
3536:
3531:
3527:
3522:
3521:
3520:
3517:
3511:
3504:
3500:
3494:
3488:
3482:
3478:
3474:
3461:
3455:
3449:
3444:
3439:
3434:
3429:
3423:
3415:
3410:
3399:
3395:
3394:
3393:
3386:
3380:
3376:
3372:
3368:
3367:
3366:
3363:
3357:
3353:
3348:
3342:
3336:
3332:
3326:
3315:
3313:
3312:
3307:
3303:
3300:Every finite
3298:
3296:
3287:
3284:
3278:
3272:
3266:
3262:
3243:
3237:
3234:
3225:
3219:
3213:
3210:
3190:
3184:
3181:
3158:
3155:
3149:
3143:
3140:
3129:
3124:
3119:
3114:
3111:
3104:
3098:
3094:
3093:abelian group
3089:
3074:
3068:
3064:
3059:
3055:
3051:
3047:
3043:
3039:
3035:
3030:
3027:
3023:
3018:
3013:
3008:
3007:
3006:
3003:
2997:
2991:
2989:
2983:
2979:
2975:
2971:
2967:
2964:
2960:
2953:
2943:
2937:
2930:
2924:
2917:
2913:
2905:
2894:
2891:
2886:
2881:
2875:
2869:
2858:
2848:
2844:
2843:
2837:
2831:
2826:
2814:
2809:
2805:
2795:
2791:
2787:
2783:
2779:
2775:
2771:
2764:
2760:
2756:
2752:
2748:
2744:
2740:
2739:
2738:
2731:
2725:
2719:
2713:
2707:
2701:
2686:
2682:
2679:
2675:
2672:
2668:
2665:
2661:
2658:
2654:
2653:
2649:
2645:
2642:
2638:
2635:
2631:
2628:
2624:
2621:
2617:
2616:
2612:
2608:
2605:
2601:
2597:
2592:
2588:
2583:
2580:
2576:
2575:
2571:
2567:
2564:
2560:
2556:
2551:
2547:
2542:
2539:
2535:
2534:
2530:
2526:
2523:
2519:
2516:
2512:
2509:
2505:
2502:
2501:
2491:
2487:
2484:
2480:
2477:
2473:
2470:
2466:
2463:
2459:
2458:
2454:
2450:
2447:
2443:
2440:
2436:
2433:
2429:
2426:
2422:
2421:
2417:
2413:
2410:
2406:
2402:
2397:
2393:
2388:
2385:
2381:
2380:
2376:
2372:
2369:
2365:
2361:
2356:
2352:
2347:
2344:
2340:
2339:
2335:
2331:
2328:
2324:
2321:
2317:
2314:
2310:
2307:
2306:
2302:
2294:
2288:
2278:
2276:
2257:
2253:
2242:
2239:
2233:
2214:
2209:
2206:
2197:
2191:
2187:line through
2186:
2185:perpendicular
2181:
2177:
2172:
2167:
2162:
2156:
2150:
2145:
2140:
2135:
2131:
2127:
2123:
2117:
2108:
2101:
2095:
2091:
2087:asserts that
2086:
2081:
2072:
2069:
2065:
2056:
2052:
2048:
2044:
2040:
2036:
2032:
2028:
2024:
2020:
2016:
2012:
2008:
2003:
2002:
2001:
1998:
1991:
1986:
1981:
1971:
1967:
1960:
1956:
1952:
1951:
1950:
1947:
1943:
1938:
1934:
1928:
1924:
1915:
1888:
1883:
1880:
1875:
1870:
1867:
1862:
1857:
1854:
1849:
1844:
1841:
1832:
1822:
1819:
1814:
1811:
1805:
1799:
1796:
1791:
1788:
1782:
1776:
1773:
1770:
1765:
1762:
1759:
1753:
1747:
1744:
1741:
1736:
1733:
1730:
1724:
1718:
1715:
1712:
1704:
1701:
1698:
1695:
1692:
1686:
1676:
1666:
1663:
1658:
1655:
1652:
1649:
1646:
1640:
1635:
1632:
1627:
1623:
1616:
1613:
1608:
1605:
1599:
1593:
1590:
1585:
1582:
1575:
1571:
1566:
1563:
1554:
1546:
1540:
1537:
1532:
1527:
1524:
1519:
1514:
1511:
1506:
1501:
1498:
1492:
1488:
1483:
1480:
1471:
1463:
1457:
1454:
1449:
1444:
1441:
1435:
1431:
1426:
1423:
1409:
1408:
1407:
1390:
1387:
1381:
1378:
1373:
1370:
1364:
1359:
1356:
1351:
1346:
1343:
1334:
1333:
1332:
1329:
1325:
1318:
1312:
1308:
1300:
1295:
1290:
1284:
1278:
1274:
1270:
1264:
1249:
1246:
1242:
1224:
1220:
1208:
1204:
1200:
1189:
1187:
1179:
1174:
1164:
1154:
1150:
1146:
1142:
1138:
1134:
1130:
1125:
1122:
1117:
1113:
1107:
1102:
1099:, called the
1097:
1090:
1086:, denoted by
1084:
1078:
1071:
1066:
1063:
1058:
1054:
1048:
1043:
1040:, called the
1038:
1031:
1025:
1019:
1014:
1011:
1007:
1003:
997:
993:
987:
974:
970:
967:
963:
959:
955:
951:
945:
941:
937:
933:
928:
927:Commutativity
925:
921:
917:
913:
909:
905:
901:
895:
891:
887:
883:
879:
875:
870:
869:Associativity
867:
866:
865:
860:of the field
859:
843:
842:
836:
832:
826:
820:
814:
808:
802:
796:
792:
786:
780:
774:
768:
754:
750:
746:
742:is a mapping
737:
733:
725:
720:
716:
702:
698:
694:
690:
686:
681:
677:
673:
669:
665:
664:
663:
660:
656:
650:
646:
641:
632:
628:
619:
614:
610:
606:
601:
597:
591:
587:
582:
572:
570:
566:
565:number theory
562:
561:Number fields
558:
554:
550:
546:
541:
539:
535:
532:that general
531:
527:
523:
522:Galois theory
519:
515:
511:
506:
504:
500:
499:finite fields
496:
492:
488:
486:
481:
477:
473:
469:
465:
461:
456:
454:
453:number theory
450:
446:
442:
438:
434:
430:
426:
422:
418:
414:
410:
398:
393:
391:
386:
384:
379:
378:
376:
375:
367:
364:
362:
359:
357:
354:
352:
349:
347:
344:
342:
339:
337:
334:
333:
329:
326:
325:
321:
316:
315:
308:
307:
303:
302:
298:
295:
293:
290:
288:
285:
284:
279:
274:
273:
266:
265:
261:
259:
256:
255:
251:
248:
246:
243:
241:
238:
236:
233:
231:
228:
226:
223:
222:
217:
212:
211:
206:
205:
198:
195:
193:
192:Division ring
190:
188:
185:
183:
180:
178:
175:
173:
170:
168:
165:
163:
160:
158:
155:
153:
150:
149:
144:
139:
138:
133:
132:
125:
122:
120:
117:
115:
114:Abelian group
112:
111:
107:
104:
102:
99:
97:
93:
90:
88:
85:
84:
80:
75:
74:
71:
68:
67:
61:
57:
53:
50:
45:
41:
37:
33:
19:
13650:
13646:
13605:
13576:
13567:
13557:
13543:
13533:
13490:
13484:
13449:
13405:
13376:
13348:Local fields
13347:
13317:
13313:
13290:
13271:
13226:
13189:
13166:(3): 73–78,
13163:
13157:
13120:
13100:
13062:
13045:, Springer,
13042:
13016:
12986:
12957:
12925:
12921:
12887:
12847:(1): 71–98,
12844:
12840:
12818:
12791:
12787:
12780:Hensel, Kurt
12765:
12744:
12732:
12729:
12709:
12679:
12664:, Springer,
12661:
12619:
12606:
12572:
12556:
12533:
12499:Local fields
12498:
12480:, Springer,
12477:
12443:, Springer,
12440:
12411:
12382:
12360:
12336:
12332:
12308:math/0105155
12298:
12292:
12265:
12261:
12218:
12214:
12184:
12163:
12145:
12124:
12113:
12102:
12096:Serre (1992)
12091:
12086:, Chapter IV
12084:Serre (1996)
12079:
12068:
12056:
12049:Artin (1991)
12044:
12038:Serre (2002)
12033:
12022:
12011:
11999:
11987:
11975:
11959:
11947:
11935:
11923:
11911:
11900:
11894:Serre (1979)
11889:
11877:
11872:, Chapter 14
11865:
11853:
11841:
11829:
11817:
11805:
11790:
11783:Artin (1991)
11778:
11766:
11759:Artin (1991)
11754:
11747:Artin (1991)
11742:
11730:
11718:
11711:Artin (1991)
11706:
11694:
11682:
11671:Corry (2004)
11666:
11654:
11638:
11627:
11620:
11613:Corry (2004)
11608:
11596:
11584:
11572:
11560:
11548:
11536:
11524:
11512:
11500:
11488:
11476:
11464:
11457:Artin (1991)
11452:
11440:
11424:
11417:Clark (1984)
11412:
11407:, p. 16
11405:McCoy (1968)
11400:
11395:, p. 10
11388:
11376:
11358:
11353:
11343:
11339:
11333:
11323:
11313:
11308:
11303:
11295:
11289:
11276:
11270:
11264:
11255:
11241:
11159:
11151:, such that
11140:
11130:
11117:
11105:are fields.
11097:
11071:
11058:
11049:
11043:
11037:
11032:
11026:
11023:
11001:
10995:
10973:
10966:
10962:
10951:
10936:
10925:
10871:
10865:
10855:
10852:
10845:
10840:
10836:
10832:
10823:
10817:
10799:
10792:
10788:
10780:
10773:
10769:
10763:
10749:
10742:
10738:
10732:
10725:
10718:
10714:
10710:
10689:
10683:
10679:
10670:
10661:
10654:
10644:
10637:
10633:
10625:
10621:
10617:
10611:
10567:
10562:
10558:
10551:
10547:
10541:
10523:
10504:
10498:
10480:
10474:
10467:
10463:
10452:
10448:
10429:
10423:
10413:
10407:
10399:
10395:
10388:
10384:
10378:
10368:
10358:
10352:
10346:
10332:
10323:
10320:
10317:
10253:
10251:
10241:
10238:
10229:
10225:
10221:
10217:
10213:
10209:
10205:
10190:
10181:
10153:
10146:
10142:
10138:
10134:
10120:
10113:
10109:
10100:
10089:
10085:
10082:
10074:
10068:
10062:
10058:
10054:
10050:
10044:
10037:
10031:
10025:
10019:
10001:
9986:
9978:vector space
9922:
9916:
9913:
9907:
9903:
9890:
9887:
9879:Applications
9870:
9863:
9861:agrees with
9856:
9849:
9839:
9835:
9828:
9814:
9705:
9554:
9538:
9534:
9530:
9518:
9512:Brauer group
9494:
9487:
9483:
9477:
9474:
9467:
9464:
9460:
9455:
9451:
9446:
9442:
9431:
9423:
9419:
9410:
9406:
9399:
9391:Prüfer group
9383:
9379:
9372:
9367:
9361:
9354:
9343:
9339:
9327:
9319:
9314:
9310:
9306:
9300:
9296:
9292:
9281:
9278:
9198:
9194:
9183:
9179:
9176:
9168:
9164:
9159:
9148:
9141:
9137:
9133:ultraproduct
9127:
9121:
9114:
9110:
9104:
9094:
9091:
9085:
9079:
9073:
9067:
9065:states that
9057:
9051:
9048:
9041:
9035:
9029:
9022:
9018:
9009:
9002:
8984:
8978:
8968:
8962:
8954:model theory
8951:
8941:
8936:
8929:
8925:
8920:
8915:
8900:
8894:
8888:
8882:
8872:
8869:
8857:
8850:
8846:
8842:
8837:
8828:
8822:
8816:
8812:
8805:
8797:
8791:
8784:
8780:
8771:
8767:
8760:
8756:
8750:
8741:
8736:
8730:
8726:
8721:
8716:
8712:
8704:
8700:
8693:
8689:
8685:
8681:
8667:
8660:
8621:
8617:
8603:
8599:
8592:
8588:
8569:
8563:
8559:
8555:
8546:
8540:
8529:
8525:
8521:Galois group
8518:
8508:
8502:
8496:
8490:
8484:
8481:
8473:
8468:
8463:
8459:
8434:
8430:
8413:
8387:
8383:
8373:
8197:
8192:
8188:
8182:
8175:
8171:
8165:
8159:
8143:
8139:
8128:
8123:
8119:
8112:
8108:
8093:
8088:
8084:
8073:
8069:
8062:
8058:
8047:
8042:
8038:
8034:
8027:
8023:
8019:
8016:
8009:
8002:
7998:
7991:
7986:
7982:
7972:
7968:
7959:local fields
7957:
7955:
7952:Local fields
7943:
7939:
7932:
7924:
7920:
7913:
7908:
7899:
7891:
7887:
7884:
7876:
7868:
7864:
7856:
7847:
7839:
7835:
7826:
7816:
7808:
7804:
7797:
7790:
7782:
7773:
7766:
7753:
7749:
7740:
7714:
7702:
7698:
7694:
7683:
7679:
7673:
7667:
7663:
7657:
7644:
7638:
7629:
7623:
7616:
7610:
7605:
7603:
7596:
7592:
7588:
7584:
7564:
7560:
7554:
7550:
7540:
7533:
7517:
7511:
7501:
7498:
7487:
7478:Dedekind cut
7476:
7473:lower bounds
7469:upper bounds
7462:
7448:
7441:
7427:
7421:
7415:
7404:
7390:
7382:
7378:
7371:
7364:
7361:
7259:
7252:
7245:
7238:
7234:
7229:
7225:
7223:
7209:
7198:
7183:
7179:
7169:
7163:
7156:
7152:
7145:
7141:
7126:
7123:. The field
7115:
7108:
7100:
7094:
7089:
7086:
7076:
7066:
7052:
7046:
7040:
7037:
7031:
7025:
7015:
7008:
7003:
7000:
6993:
6985:
6981:
6976:
6966:
6962:
6959:
6951:
6947:
6940:
6935:
6931:
6921:
6917:
6911:
6907:
6902:
6898:
6894:
6889:
6885:
6872:
6857:
6853:
6849:
6843:
6839:
6832:
6828:
6822:
6816:
6802:
6796:
6793:
6787:
6781:
6774:
6770:
6763:
6759:
6753:
6743:
6737:
6731:
6727:
6723:
6720:
6714:
6708:
6698:
6692:
6686:
6680:
6670:
6663:
6659:
6653:
6642:
6638:
6634:
6628:
6624:
6620:
6614:
6608:
6602:
6596:
6592:
6586:
6575:
6571:
6568:
6466:
6462:
6456:
6450:
6444:
6438:
6431:
6427:
6421:
6417:
6413:
6406:
6402:
6396:
6390:
6384:
6378:
6371:
6367:
6364:
6354:
6348:
6345:
6337:
6328:
6322:
6316:
6310:
6299:
6295:
6282:
6277:
6273:
6270:
6259:
6255:
6249:
6245:
6240:
6236:
6232:
6227:
6223:
6209:coefficients
6192:
6187:
6183:
6173:
6169:
6166:
6155:
6151:
6138:
6131:
6125:
6121:
6118:
6104:
6098:
6091:
6085:
6079:
6076:
6069:
6065:
6056:
6049:
6044:
6038:
6034:
6031:
6014:
6008:
6002:
5993:
5987:
5981:
5975:
5966:
5960:
5953:
5946:
5942:
5936:
5930:
5924:
5918:
5914:
5908:
5900:
5894:
5890:
5884:
5877:
5873:
5871:and denoted
5867:
5862:
5857:
5851:
5845:
5839:
5833:
5827:
5821:
5815:
5812:
5705:
5700:
5693:
5689:
5684:
5677:
5671:
5669:. Moreover,
5664:
5660:
5655:
5647:
5642:
5623:
5617:
5614:
5606:
5601:
5591:
5585:(namely the
5581:
5575:
5572:
5565:
5560:
5555:
5551:
5538:
5532:
5526:
5516:
5510:
5504:
5500:
5494:
5488:
5482:
5472:
5464:
5458:
5448:
5441:
5437:
5427:
5417:
5408:
5405:
5394:
5387:
5376:
5370:
5367:
5257:
5253:
5251:. The field
5247:
5238:
5227:
5223:
5220:
5217:
5133:
5129:
5123:
5119:
5113:
5109:
5102:
5096:
5090:
5084:
5078:
5074:
5067:
5063:
5058:
5053:
5046:
5042:
5033:
5027:
5015:
5011:
5005:
4999:
4993:
4987:
4981:
4975:
4972:
4966:is the only
4958:
4940:
4937:
4930:
4924:
4914:
4908:
4901:
4852:
4837:
4833:
4825:Weber (1893)
4822:
4816:
4810:
4791:
4787:
4780:
4770:
4764:
4753:
4748:Moore (1893)
4743:
4733:
4698:
4690:
4684:Galois group
4678:
4676:for a prime
4675:
4668:
4658:
4641:
4635:
4617:
4614:
4604:
4601:
4594:
4587:
4570:
4563:
4556:
4541:
4529:
4521:
4517:
4511:
4506:
4500:
4486:
4477:
4471:
4465:
4461:
4452:
4445:
4439:
4433:
4423:
4417:
4411:
4404:
4400:
4397:
4390:
4385:
4380:
4375:
4358:
4354:
4351:
4345:
4337:vector space
4331:
4324:
4318:
4312:
4308:
4302:
4299:
4292:
4288:
4282:
4276:
4270:
4267:
4263:
4257:
4254:
4250:
4240:
4237:
4233:
4226:
4222:
4216:
4213:
4209:
4203:
4199:
4195:
4189:
4178:
4169:
4162:
4158:prime number
4152:
4146:
4140:
4137:
4130:
4126:
4123:
4119:
4110:
4104:
4097:
4086:
4082:
4068:
4036:
4027:
4022:
4018:
4017:
4014:Finite field
4000:
3993:
3989:
3983:
3977:
3971:
3964:
3958:
3952:
3938:
3928:
3919:
3910:
3903:
3897:
3892:
3882:
3878:
3871:
3867:
3860:
3854:
3850:
3840:
3836:
3829:
3825:
3818:
3811:
3807:
3801:
3797:
3793:
3787:
3781:
3775:
3768:
3761:
3754:
3748:
3742:
3738:
3732:
3728:
3722:
3718:
3714:
3708:
3697:
3691:
3685:
3679:
3673:
3667:
3661:
3659:
3644:
3638:
3635:
3628:
3623:
3619:
3615:
3603:
3597:
3591:
3585:
3579:
3575:
3571:
3566:
3559:
3548:
3540:
3534:
3529:
3525:
3515:
3509:
3502:
3498:
3492:
3486:
3483:
3476:
3472:
3459:
3453:
3447:
3443:prime number
3437:
3432:
3427:
3421:
3408:
3405:
3397:
3391:
3384:
3378:
3374:
3370:
3361:
3355:
3346:
3340:
3334:
3330:
3324:
3321:
3309:
3299:
3295:trivial ring
3288:
3282:
3276:
3270:
3261:distributive
3223:
3217:
3214:
3208:
3122:
3120:elements of
3117:
3115:
3109:
3102:
3087:
3084:
3072:
3066:
3062:
3053:
3049:
3045:
3041:
3037:
3033:
3025:
3021:
3001:
2995:
2992:
2981:
2977:
2973:
2969:
2965:
2962:
2958:
2951:
2949:, since, if
2941:
2935:
2928:
2925:
2915:
2911:
2903:
2900:
2889:
2879:
2873:
2867:
2864:
2846:
2842:binary field
2840:
2835:
2829:
2812:
2808:Galois field
2807:
2804:finite field
2801:
2793:
2789:
2785:
2781:
2777:
2773:
2769:
2762:
2758:
2754:
2750:
2746:
2742:
2729:
2723:
2717:
2711:
2705:
2699:
2696:
2684:
2677:
2670:
2663:
2656:
2647:
2640:
2633:
2626:
2619:
2610:
2603:
2595:
2586:
2578:
2569:
2562:
2554:
2545:
2537:
2528:
2521:
2514:
2507:
2489:
2482:
2475:
2468:
2461:
2452:
2445:
2438:
2431:
2424:
2415:
2408:
2400:
2391:
2383:
2374:
2367:
2359:
2350:
2342:
2333:
2326:
2319:
2312:
2243:
2237:
2231:
2195:
2189:
2179:
2170:
2160:
2154:
2148:
2144:square roots
2138:
2134:straightedge
2119:
2106:
2099:
2093:
2089:
2061:
2054:
2050:
2046:
2042:
2038:
2034:
2030:
2026:
2022:
2018:
2014:
2010:
2006:
1996:
1989:
1979:
1976:
1969:
1965:
1958:
1954:
1945:
1936:
1933:real numbers
1930:
1405:
1327:
1323:
1316:
1310:
1306:
1298:
1288:
1282:
1276:
1272:
1266:
1244:
1240:
1235:0 = 1 + (−1)
1195:
1175:
1160:
1152:
1148:
1144:
1140:
1136:
1132:
1128:
1115:
1111:
1109:, such that
1105:
1100:
1095:
1088:
1082:
1076:
1069:
1067:: for every
1056:
1052:
1050:, such that
1046:
1041:
1036:
1029:
1023:
1017:
1015:: for every
1005:
1001:
995:
991:
985:
961:
957:
953:
949:
943:
939:
935:
931:
919:
915:
911:
907:
903:
899:
893:
889:
885:
881:
877:
873:
841:field axioms
839:
834:
830:
824:
818:
812:
806:
800:
794:
790:
784:
778:
772:
766:
752:
748:
744:
735:
731:
718:
712:
700:
696:
692:
688:
679:
675:
671:
667:
658:
654:
648:
644:
639:
630:
617:
609:real numbers
599:
595:
589:
585:
578:
553:vector space
542:
507:
487:-adic fields
484:
464:real numbers
457:
441:real numbers
412:
406:
366:Hopf algebra
304:
297:Vector space
262:
202:
186:
131:Group theory
129:
94: /
40:
32:Vector field
13410:Patrick Ion
12983:Lang, Serge
12221:: 225–231,
12207:Artin, Emil
12129:Baez (2002)
12107:Tits (1957)
11966:. See also
11952:Lang (2002)
11940:Lang (2002)
11928:Lang (2002)
11699:Lang (2002)
11603:, pp. 75–76
11419:, Chapter 3
11093:John Milnor
11083:in 1958 by
11077:alternative
11055:quaternions
11014:game theory
10987:near-fields
10983:quasifields
10432:isomorphism
9920:in a field
9895:, then the
9847:shows that
9824:determinant
9338:exp :
9100:ultrafilter
8990:first-order
8972:are called
8910:cardinality
8906:uncountable
8550:(i.e., the
8104:first-order
8054:uniformizer
7851:any field)
7719:) is zero.
6873:A field is
6800:of a field
6600:where both
6199:if it is a
5928:containing
5849:containing
5825:containing
5573:This field
5542:of smaller
5530:, i.e., if
5522:irreducible
5413:injectively
5057:exactly as
4968:prime ideal
4804:(1873) and
4651:Vandermonde
4248:, prevents
4080:. However,
3967:prime field
3948:isomorphism
3671:of a field
2193:in a point
2097:. Choosing
1923:Real number
625:, and of a
425:subtraction
409:mathematics
351:Lie algebra
336:Associative
240:Total order
230:Semilattice
204:Ring theory
13699:Categories
13675:25.0137.01
13595:0683.12014
13515:41.0445.03
13470:0674.13008
13440:1004.12003
13395:0746.12001
13337:0432.10001
13103:, Boston:
13035:1139.11053
12808:35.0227.01
12719:50.0042.03
12591:1044.01008
12569:Corry, Leo
12430:0978.12004
12353:0739.03027
12243:53.0144.01
12137:References
11089:Raoul Bott
11033:skew field
10991:semifields
10934:describes
9500:Weil group
8832:-algebras
8820:of degree
8552:bijections
8379:derivation
7885:The field
7731:Completion
7642:, such as
7619:completion
7576:, i.e., a
7514:hyperreals
7087:Any field
6205:polynomial
6149:, whereas
6077:and read "
5956:compositum
5904:an element
5385:(in which
5221:The field
4920:reciprocal
4883:Emil Artin
4368:polynomial
4280:elements (
4064:9 + 4 = 13
3944:surjective
3365:-fold sum
3359:to be the
2166:semicircle
1186:invertible
1033:, denoted
989:such that
581:operations
575:Definition
536:cannot be
13683:120528969
13667:0025-5831
13523:120807300
13507:0075-4102
13310:(1996) ,
13069:CiteSeerX
12950:121442989
12877:119378923
12800:0075-4102
12772:EMS Press
12258:Ax, James
12251:121547404
12235:0025-5858
11369:Citations
11206:−
11196:⋅
11181:∖
11068:octonions
10977:tends to
10903:−
10444:dimension
10183:Functions
9945:−
9926:, namely
9789:⊗
9780:μ
9668:∖
9662:∈
9656:∣
9647:−
9638:⊗
9620:×
9612:⊗
9609:⋯
9606:⊗
9601:×
9522:-algebras
9256:≅
9245:¯
9234:
9229:∞
9226:→
9102:on a set
9083:holds in
8583:subgroups
8500:and that
8440:separable
8338:∞
8281:
8275:≅
8259:∞
8217:
7400:Witt ring
7398:from the
7323:⋯
7250:whenever
7021:algebraic
7019:if it is
6970:. By the
6810:if it is
6794:A subset
6747:is not a
6548:∈
6501:−
6483:∑
6193:algebraic
5902:adjoining
5778:≅
5773:⟶
5629:adjoining
5347:∈
5326:∈
5292:∞
5277:∑
5028:Given an
4719:in 1824.
4657:, in his
4341:dimension
4060:9 + 4 = 1
3934:injective
3791:are maps
3570: :=
3238:∖
3185:∖
3159:⋅
3144:∖
2914:= (−1) ⋅
2296:Addition
1876:⋅
1850:⋅
1641:⋅
1572:⋅
1533:⋅
1507:⋅
1489:⋅
1432:⋅
1352:⋅
1269:fractions
1223:computing
695: :=
674: :=
419:on which
361:Bialgebra
167:Near-ring
124:Lie group
92:Semigroup
13641:(1893),
13604:(1997),
13555:(1857),
13479:(1910),
13404:(2002),
13375:(1992),
13346:(1979),
13262:(2014),
13225:(1999),
13113:68015225
13015:(2008),
12985:(2002),
12817:(2009),
12794:: 1–32,
12782:(1904),
12743:(2012),
12728:(1997),
12707:(1924),
12618:(1995),
12600:(1871),
12571:(2004),
12554:(1976),
12497:(1986),
12476:(1988),
12439:(1994),
12381:(1985),
12183:(1991),
11737:, p. 213
11519:, 14.4.2
11215:⟩
11174:⟨
11158:⟨
11145:of type
11047:itself,
10693:being a
10676:for all
10631:, where
9897:equation
9687:⟩
9631:⟨
9551:K-theory
8609:solvable
8558: :
8420:symmetry
8080:carrying
7931:complex
7710:sequence
7606:distance
7587: :
7578:function
7507:calculus
7224:A field
6757:), then
6668:, where
6048:being a
5652:, where
5446:, where
5433:quotient
4765:In 1881
4758:—
4734:In 1871
4688:regular
3942:is also
3908:, where
3877:)
3663:subfield
3622: :
3563:. Here,
3507:for all
3377:+ ... +
3302:subgroup
3174:or just
2945:must be
2901:One has
2885:elements
2176:midpoint
2164:, and a
1294:integers
1280:, where
1252:Examples
1233:, since
969:Additive
858:elements
732:addition
503:elements
497:rely on
437:rational
433:division
421:addition
197:Lie ring
162:Semiring
52:heptagon
13632:1421575
13432:1867431
13366:0554237
13253:1677964
13216:0769847
13182:1557275
13147:0919949
13093:2215060
12987:Algebra
12976:1760173
12942:1554154
12914:2347309
12869:0679774
12849:Bibcode
12774:, 2001
12762:"Field"
12654:1322960
12604:(ed.),
12525:0861410
12467:1290116
12282:1970573
12185:Algebra
11942:, §VI.1
11785:, §13.9
11725:, p. 60
11713:, §10.6
11701:, §II.1
11689:, p. 92
11661:, p. 81
11649:, p. 66
11615:, p. 24
11591:, p. 50
11579:, p. 63
11543:, p. 12
11495:, Th. 2
11253:within
11010:nimbers
10678:0 <
10494:compact
10394:, ...,
9989:modules
9135:of the
8766:, ...,
8748:(where
7759:(usual
7649:√
7467:if all
6996:+ 1 = 0
6939:, ...,
6893:
6820:and if
6340:+ 1 = 0
6281:, ...,
6231:
5635:symbol
5232:of the
5094:are in
4755:system.
4701:= 2 + 1
4538:History
4448:= 2 = 4
4366:of the
3779:are in
3765:, both
3746:are in
3496:, then
3400:⋅ 1 = 0
3352:integer
3118:nonzero
2984:⋅ 0 = 0
2956:, then
2906:⋅ 0 = 0
2130:compass
2066:, with
1983:is the
1207:nullary
730:called
640:inverse
549:scalars
493:. Most
449:algebra
328:Algebra
320:Algebra
225:Lattice
216:Lattice
49:regular
13681:
13673:
13665:
13630:
13620:
13593:
13583:
13521:
13513:
13505:
13468:
13458:
13438:
13430:
13420:
13393:
13383:
13364:
13354:
13335:
13325:
13297:
13279:
13251:
13241:
13214:
13204:
13180:
13145:
13135:
13111:
13091:
13081:
13071:
13049:
13033:
13023:
13001:
12974:
12964:
12948:
12940:
12912:
12902:
12875:
12867:
12825:
12806:
12798:
12751:
12717:
12690:
12668:
12652:
12642:
12589:
12579:
12541:
12523:
12513:
12484:
12465:
12455:
12428:
12418:
12402:
12394:
12367:
12351:
12325:586512
12323:
12280:
12249:
12241:
12233:
12195:
12170:
12152:
12051:, §3.3
11884:, §5.7
11531:, §I.3
11091:, and
11066:. The
10940:using
10928:> 0
10875:. For
10699:Kummer
10687:. For
10512:. The
10490:proper
10486:smooth
10459:curves
10061:⋅ ⋯ ⋅
10029:, and
9980:has a
9533:) = H(
9434:> 0
9393:, the
9304:≅ ulim
9131:, the
9108:, and
9098:is an
9005:> 0
8740:+ ⋯ +
8634:
8622:cannot
8482:where
8452:simple
8446:. The
8444:normal
8424:finite
7757:|
7747:|
7728:Metric
7706:|
7692:|
7574:metric
7568:) are
7377:= ⋯ =
7268:. The
7064:, and
6954:> 0
6910:+ ⋯ +
6674:is an
6535:
6532:
6436:as an
6293:, and
6248:+ ⋯ +
6102:as an
6094:degree
5785:
5766:
5544:degree
5317:
5100:, and
5082:where
5037:, its
4950:ideals
4744:Körper
4740:German
4615:(with
4546:, and
4498:), so
4443:. For
4429:degree
4176:. For
4062:since
3890:, and
3578:⋅ ⋯ ⋅
3306:cyclic
3265:groups
3048:) = −(
3010:−0 = 0
2825:subset
2823:. The
2715:, and
1977:where
1296:, and
1243:= (−1)
1180:where
1004:⋅ 1 =
994:+ 0 =
947:, and
897:, and
852:, and
551:for a
482:, and
431:, and
356:Graded
287:Module
278:Module
177:Domain
96:Monoid
13679:S2CID
13519:S2CID
13267:(PDF)
12946:S2CID
12873:S2CID
12321:S2CID
12303:arXiv
12278:JSTOR
12264:, 2,
12247:S2CID
11293:Both
11109:Notes
11002:Field
10971:, as
10954:0 ≠ 1
10682:<
10478:: if
10374:space
10327:is a
10254:field
10176:genus
9993:rings
9982:basis
8698:(and
7725:Field
7544:is a
7475:(see
7419:, to
7413:over
7023:over
6806:is a
6271:with
6207:with
6203:of a
6195:over
6083:over
5650:) = 0
5609:) = 0
5462:. If
5452:is a
4742:word
4577:of a
4274:with
4246:above
4156:is a
4133:− 1}.
3936:. If
3901:) = 1
3726:both
3406:then
3308:(see
3291:1 ≠ 0
3206:, or
3024:)) =
3015:1 = 1
2980:) =
2933:then
2855:GF(2)
2821:GF(4)
2235:when
2229:from
2168:over
2064:plane
2045:) + (
1973:real,
1963:with
1321:) is
1182:0 ≠ 1
1163:group
1147:) + (
1139:) = (
1059:) = 0
910:) = (
884:) = (
415:is a
413:field
322:-like
280:-like
218:-like
187:Field
145:-like
119:Magma
87:Group
81:-like
79:Group
13663:ISSN
13618:ISBN
13581:ISBN
13503:ISSN
13491:1910
13456:ISBN
13418:ISBN
13381:ISBN
13352:ISBN
13323:ISBN
13295:ISBN
13277:ISBN
13239:ISBN
13202:ISBN
13133:ISBN
13109:LCCN
13079:ISBN
13047:ISBN
13021:ISBN
12999:ISBN
12962:ISBN
12900:ISBN
12823:ISBN
12796:ISSN
12749:ISBN
12688:ISBN
12666:ISBN
12640:ISBN
12577:ISBN
12539:ISBN
12511:ISBN
12482:ISBN
12453:ISBN
12416:ISBN
12400:ISBN
12392:ISBN
12365:ISBN
12231:ISSN
12193:ISBN
12168:ISBN
12150:ISBN
12006:, §2
11918:, §1
11301:ulim
11299:and
11274:and
11165:and
10989:and
10848:≥ 2)
10787:Gal(
10767:and
10747:and
10667:and
10531:and
10488:and
10434:and
10419:open
10353:The
10224:) ⋅
10216:) =
10158:and
9838:) =
9706:The
9459:) =
9441:Gal(
9429:for
9418:Gal(
9405:Gal(
9378:Gal(
9360:Gal(
9290:ulim
9219:ulim
9157:ulim
8966:and
8934:and
8898:and
8806:The
8719:) =
8688:) =
8587:Gal(
8524:Gal(
8442:and
8186:(in
8169:(in
8102:Any
8032:and
7677:and
7617:The
7558:and
7512:The
7257:and
7243:and
6856:) /
6749:root
6730:) /
6606:and
6420:) /
6201:root
6177:are
6129:and
5991:and
5964:and
5954:The
5934:and
5855:and
5692:) ∊
5663:) =
5631:the
5489:The
5117:and
5088:and
4956:and
4946:unit
4933:= ±1
4869:and
4814:and
4694:-gon
4629:and
4494:and
4415:has
4383:) =
4306:has
4047:and
3956:and
3917:and
3866:) =
3835:) +
3824:) =
3772:and
3736:and
3707:1 ∊
3532:) =
3274:and
3105:, +)
3065:) =
3044:⋅ (−
3036:) ⋅
3020:(−(−
2999:and
2908:and
2877:and
2833:and
2753:) =
2132:and
2083:The
2021:) =
1992:= −1
1931:The
1925:and
1292:are
1286:and
1237:and
1229:and
1221:and
1213:and
1055:+ (−
999:and
979:and
971:and
918:) ⋅
892:) +
816:and
804:and
782:and
770:and
734:and
678:+ (−
607:and
512:and
439:and
411:, a
152:Ring
143:Ring
47:The
13671:JFM
13655:doi
13610:doi
13591:Zbl
13511:JFM
13495:doi
13466:Zbl
13436:Zbl
13391:Zbl
13333:Zbl
13231:doi
13194:doi
13168:doi
13125:doi
13031:Zbl
12991:doi
12930:doi
12892:doi
12857:doi
12804:JFM
12792:128
12715:JFM
12632:doi
12587:Zbl
12503:doi
12445:doi
12426:Zbl
12349:Zbl
12341:doi
12313:doi
12270:doi
12239:JFM
12223:doi
11095:.
11031:or
10674:≠ 1
10665:= 1
10648:th
10597:).
10496:),
10484:is
10382:is
10344:on
10121:In
10077:≥ 1
9893:≠ 0
9888:If
9529:Br(
9492:).
9397:of
9368:all
9347:).
9332:or
9125:in
9092:If
9033:in
8952:In
8942:not
8880:of
8800:≥ 5
8696:+ 2
8692:− 4
8670:≥ 5
8585:of
8538:of
8278:Gal
8214:Gal
7717:→ ∞
7621:of
7553:↦ −
7485:of
7428:An
7409:of
7386:= 0
7262:≥ 0
7255:≥ 0
7248:≥ 0
7241:≥ 0
7188:of
7113:of
7095:the
7074:to
7070:is
7013:of
6986:not
6982:any
6927:= 0
6785:to
6580:of
6314:in
6303:≠ 0
6289:in
6265:= 0
6211:in
6191:is
6113:=
6089:".
5906:to
5888:to
5837:of
5697:to
5667:+ 1
5627:by
5589:of
5568:)).
5558:/ (
5524:in
5520:is
5480:of
5456:of
5390:≥ 0
5381:of
5262:of
5105:≠ 0
4997:is
4964:(0)
4954:(0)
4783:(π)
4671:= 1
4528:GF(
4526:or
4507:the
4437:is
4431:of
4327:≥ 1
4231:in
4229:= 0
4181:= 4
4165:= 2
4085:/12
4066:in
3981:is
3759:in
3757:≠ 0
3513:in
3505:= 0
3484:If
3479:= O
3344:of
3314:).
3075:≠ 0
3070:if
2961:= (
2954:≠ 0
2939:or
2931:= 0
2926:If
2887:of
2853:or
2819:or
2806:or
2745:⋅ (
2102:= 1
2037:= (
2035:bdi
2031:adi
2027:bci
1319:≠ 0
1301:≠ 0
1131:⋅ (
1118:= 1
1103:of
1092:or
1074:in
1072:≠ 0
1044:of
1021:in
983:in
902:⋅ (
876:+ (
864:):
828:or
726:on
715:set
520:.
417:set
407:In
157:Rng
13701::
13677:,
13669:,
13661:,
13651:43
13645:,
13628:MR
13626:,
13616:,
13589:,
13517:,
13509:,
13501:,
13489:,
13464:,
13434:,
13428:MR
13426:,
13416:,
13389:,
13362:MR
13360:,
13331:,
13269:,
13249:MR
13247:,
13237:,
13212:MR
13210:,
13200:,
13178:MR
13176:,
13162:,
13143:MR
13141:,
13131:,
13107:,
13089:MR
13087:,
13077:,
13029:,
12997:,
12972:MR
12970:,
12944:,
12938:MR
12936:,
12924:,
12910:MR
12908:,
12898:,
12871:,
12865:MR
12863:,
12855:,
12845:70
12843:,
12839:,
12802:,
12786:,
12770:,
12764:,
12703:;
12686:,
12650:MR
12648:,
12638:,
12630:,
12622:,
12585:,
12560:,
12521:MR
12519:,
12509:,
12463:MR
12461:,
12451:,
12424:,
12406:).
12385:,
12347:,
12337:38
12335:,
12319:,
12311:,
12299:39
12297:,
12276:,
12266:88
12245:,
12237:,
12229:,
12209:;
12191:,
12187:,
11797:.
11087:,
10993:.
10985:,
10923:,
10879:,
10844:,
10803:.
10717:=
10713:+
10697:,
10350:.
10249:.
10212:)(
10208:⋅
10162:.
10145:+
10143:ax
10141:+
10137:=
10112:=
10057:⋅
10053:=
10023:,
9984:.
9906:=
9904:ax
9843:.
9537:,
9450:/
9342:→
9322:))
9318:((
9285:)
9188::
9162:→∞
9007:,
8996:,
8924:,
8849:≅
8815:/
8802:).
8789:,
8774:−1
8733:−1
8725:+
8708:),
8703:=
8676::
8562:→
8516:.
8466:/
8462:=
8433:/
8400:.
8200:))
8196:((
8131:))
8127:((
8096:))
8092:((
8050:))
8046:((
8037:∈
8022:∈
8013:).
7994:))
7990:((
7962::
7948:.
7871:))
7867:((
7822:)
7772:1/
7752:−
7697:−
7647:=
7614:.
7595:→
7591:×
7563:↦
7526:.
7471:,
7452:.
7425:.
7403:W(
7370:=
7246:xy
7237:+
7202:.
7186:))
7182:((
7084:.
6974:,
6965:∊
6950:,
6946:∈
6920:+
6905:−1
6897:+
6842:/
6646:.
6643:bi
6641:+
6637:−
6629:ci
6627:+
6625:bi
6623:+
6597:bi
6595:+
6258:+
6243:−1
6235:+
6186:∈
6172:/
6154:/
6137:/
6124:/
6068:/
6037:⊂
6018:.
5951:.
5917:⊂
5554:=
5503:=
5486:.
5440:/
5260:))
5256:((
5134:bc
5132:=
5130:ad
5020:.
4970:.
4952:,
4935:.
4902:A
4889:.
4865:,
4842:.
4750:.
4600:+
4595:ωx
4593:+
4569:,
4562:,
4534:.
4484:,
4464:=
4389:−
4357:=
4311:=
4297:.
4225:⋅
4202:⋅
4198:=
4092:12
4074:12
4051:.
4004:.
3895:(1
3848:,
3817:+
3800:→
3796::
3785:.
3774:1/
3741:⋅
3731:+
3721:∊
3717:,
3660:A
3652:.
3626:↦
3618:→
3574:⋅
3538:+
3528:+
3501:⋅
3481:.
3475:+
3433:is
3388:.)
3373:+
3333:⋅
3286:.
3212:.
3052:⋅
3040:=
3032:(−
3005::
2990:.
2978:ab
2972:=
2929:ab
2923:.
2921:−1
2893:.
2857:.
2792:=
2788:+
2784:=
2780:⋅
2776:+
2772:⋅
2761:=
2757:⋅
2749:+
2709:,
2703:,
2238:BD
2171:AD
2161:BD
2158:,
2155:AB
2094:pq
2092:=
2051:ad
2049:+
2047:bc
2043:bd
2041:−
2039:ac
2033:+
2029:+
2025:+
2023:ac
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2432:A
2425:A
2416:A
2409:B
2401:O
2392:I
2384:I
2375:B
2368:A
2360:I
2351:O
2343:O
2334:B
2327:A
2320:I
2313:O
2308:+
2258:3
2254:2
2232:B
2215:p
2210:=
2207:h
2196:F
2190:B
2180:C
2149:Q
2139:Q
2110:.
2107:p
2100:q
2090:h
2057:.
2055:i
2053:)
2015:c
2007:a
2005:(
1997:C
1990:i
1980:i
1970:b
1966:a
1961:,
1955:a
1946:C
1937:R
1889:.
1884:f
1881:e
1871:b
1868:a
1863:+
1858:d
1855:c
1845:b
1842:a
1833:=
1823:f
1820:b
1815:e
1812:a
1806:+
1800:d
1797:b
1792:c
1789:a
1783:=
1777:f
1774:d
1771:b
1766:d
1763:e
1760:a
1754:+
1748:f
1745:d
1742:b
1737:f
1734:c
1731:a
1725:=
1719:f
1716:d
1713:b
1708:)
1705:d
1702:e
1699:+
1696:f
1693:c
1690:(
1687:a
1677:=
1667:f
1664:d
1659:d
1656:e
1653:+
1650:f
1647:c
1636:b
1633:a
1628:=
1624:)
1617:d
1614:f
1609:d
1606:e
1600:+
1594:f
1591:d
1586:f
1583:c
1576:(
1567:b
1564:a
1555:=
1547:)
1541:d
1538:d
1528:f
1525:e
1520:+
1515:f
1512:f
1502:d
1499:c
1493:(
1484:b
1481:a
1472:=
1464:)
1458:f
1455:e
1450:+
1445:d
1442:c
1436:(
1427:b
1424:a
1388:=
1382:b
1379:a
1374:a
1371:b
1365:=
1360:b
1357:a
1347:a
1344:b
1328:a
1326:/
1324:b
1317:a
1311:b
1309:/
1307:a
1305:−
1299:b
1289:b
1283:a
1277:b
1275:/
1273:a
1245:a
1241:a
1239:−
1227:1
1215:1
1211:0
1171:1
1167:0
1157:.
1155:)
1153:c
1149:a
1145:b
1141:a
1137:c
1133:b
1129:a
1120:.
1116:a
1112:a
1106:a
1096:a
1089:a
1083:F
1077:F
1070:a
1061:.
1057:a
1053:a
1047:a
1037:a
1035:−
1030:F
1024:F
1018:a
1009:.
1006:a
1002:a
996:a
992:a
986:F
981:1
977:0
965:.
962:a
958:b
954:b
950:a
944:a
940:b
936:b
932:a
923:.
920:c
916:b
912:a
908:c
904:b
900:a
894:c
890:b
886:a
882:c
878:b
874:a
862:F
854:c
850:b
846:a
835:b
831:a
819:b
813:a
807:b
801:a
795:b
791:a
785:b
779:a
773:b
767:a
762:F
758:F
753:F
749:F
745:F
740:F
728:F
719:F
704:.
701:b
697:a
693:b
689:a
684:,
682:)
680:b
676:a
672:b
668:a
659:b
655:a
649:b
645:a
636:b
631:b
623:a
618:a
616:−
600:b
596:a
590:b
586:a
485:p
396:e
389:t
382:v
62:.
38:.
20:)
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