32:
4682:
5502:
4177:
4917:
3950:
4677:{\displaystyle {\begin{aligned}-(a+b\alpha +c\alpha ^{2})&=-a+(-b)\alpha +(-c)\alpha ^{2}\qquad {\text{(for }}\mathrm {GF} (8),{\text{this operation is the identity)}}\\(a+b\alpha +c\alpha ^{2})+(d+e\alpha +f\alpha ^{2})&=(a+d)+(b+e)\alpha +(c+f)\alpha ^{2}\\(a+b\alpha +c\alpha ^{2})(d+e\alpha +f\alpha ^{2})&=(ad+bf+ce)+(ae+bd+bf+ce+cf)\alpha +(af+be+cd+cf)\alpha ^{2}\end{aligned}}}
5497:{\displaystyle {\begin{aligned}(a+b\alpha +c\alpha ^{2}+d\alpha ^{3})+(e+f\alpha +g\alpha ^{2}+h\alpha ^{3})&=(a+e)+(b+f)\alpha +(c+g)\alpha ^{2}+(d+h)\alpha ^{3}\\(a+b\alpha +c\alpha ^{2}+d\alpha ^{3})(e+f\alpha +g\alpha ^{2}+h\alpha ^{3})&=(ae+bh+cg+df)+(af+be+bh+cg+df+ch+dg)\alpha \;+\\&\quad \;(ag+bf+ce+ch+dg+dh)\alpha ^{2}+(ah+bg+cf+de+dh)\alpha ^{3}\end{aligned}}}
3550:
7133:
3945:{\displaystyle {\begin{aligned}-(a+b\alpha )&=-a+(-b)\alpha \\(a+b\alpha )+(c+d\alpha )&=(a+c)+(b+d)\alpha \\(a+b\alpha )(c+d\alpha )&=(ac+rbd)+(ad+bc)\alpha \\(a+b\alpha )^{-1}&=a(a^{2}-rb^{2})^{-1}+(-b)(a^{2}-rb^{2})^{-1}\alpha \end{aligned}}}
8231:
7567:
8928:
6832:
8677:
an isomorphism, as do all algebraic closures, but contrarily to the general case, all its subfield are fixed by all its automorphisms, and it is also the algebraic closure of all finite fields of the same characteristic
8825:
8045:
6812:
5998:
over medium-sized fields, that is, fields that are sufficiently large for making natural algorithms inefficient, but not too large, as one has to pre-compute a table of the same size as the order of the field.
2503:
1227:
8114:
8567:
4807:
1915:
4922:
4182:
3555:
1073:
7373:
8975:
8639:
1705:
7666:
There are efficient algorithms for testing polynomial irreducibility and factoring polynomials over finite field. They are a key step for factoring polynomials over the integers or the
811:
4079:
457:
on which the operations of multiplication, addition, subtraction and division are defined and satisfy certain basic rules. The most common examples of finite fields are given by the
7252:
3213:
9058:
4146:
9343:
9132:
4882:
9197:
8719:
8671:
3117:
A table for subtraction is not given, because subtraction is identical to addition, as is the case for every field of characteristic 2. In the third table, for the division of
2274:
9277:
682:
7487:
6670:
8856:
8762:
7436:
3501:
2135:
9093:
6362:
has several interesting properties that smaller fields do not share: it has two subfields such that neither is contained in the other; not all generators (elements with
4728:
3996:
2400:
1393:
1619:
8851:
7713:
9016:
9304:
9229:
9161:
9136:
For explicit computations, it may be useful to have a coherent choice of the primitive elements for all finite fields; that is, to choose the primitive element
7663:, every monic polynomial over a finite field may be factored in a unique way (up to the order of the factors) into a product of irreducible monic polynomials.
552:; this means that multiplication, addition, subtraction and division (excluding division by zero) are defined and satisfy the rules of arithmetic known as the
9367:
over a finite field has a non-trivial zero whose components are in the field if the number of its variables is more than its degree. This was a conjecture of
840:
of the result of the corresponding integer operation. The multiplicative inverse of an element may be computed by using the extended
Euclidean algorithm (see
1312:
A more general algebraic structure that satisfies all the other axioms of a field, but whose multiplication is not required to be commutative, is called a
7953:
6682:
8386:
Similarly many theoretical problems in number theory can be solved by considering their reductions modulo some or all prime numbers. See, for example,
2409:
8767:
7155:. In fact, this generator is a primitive element, and this polynomial is the irreducible polynomial that produces the easiest Euclidean division.
7615:
1151:
9495:
9491:
416:
8500:
9715:
1990:
7128:{\displaystyle (X^{6}+X^{4}+X^{3}+X+1)(X^{6}+X+1)(X^{6}+X^{5}+1)(X^{6}+X^{5}+X^{3}+X^{2}+1)(X^{6}+X^{5}+X^{2}+X+1)(X^{6}+X^{5}+X^{4}+X+1).}
4748:
1843:
8397:
7322:
5925:
are represented by their discrete logarithms, multiplication and division are easy, as they reduce to addition and subtraction modulo
8459:
is a generalization of field. Division rings are not assumed to be commutative. There are no non-commutative finite division rings:
5904:, there is no known efficient algorithm for computing the inverse operation, the discrete logarithm. This has been used in various
1640:
8289:
protocol. For example, in 2014, a secure internet connection to
Knowledge involved the elliptic curve Diffie–Hellman protocol (
6468:
in the sense that no other subfield contains any of them. It follows that they are roots of irreducible polynomials of degree
4036:
9791:
9773:
9755:
9679:
7218:
3179:
4112:
842:
9452:
Mathematical Papers Read at the
International Mathematics Congress Held in Connection with the World's Columbian Exposition
5583:
4848:
1013:
655:
with the same order. One may therefore identify all finite fields with the same order, and they are unambiguously denoted
409:
2229:. They ensure a certain compatibility between the representation of a field and the representations of its subfields.
9821:
9592:
9550:
9348:
8941:
8313:
6363:
2232:
In the next sections, we will show how the general construction method outlined above works for small finite fields.
2226:
2005:
may be difficult to distinguish from the corresponding polynomials. Therefore, it is common to give a name, commonly
1971:
1298:
75:
53:
8608:
6626:
46:
9360:
8986:
7896:
does not have any multiple factor, it is thus the product of all the irreducible monic polynomials that divide it.
5726:
5511:
744:
8226:{\displaystyle N(q,n)\geq {\frac {1}{n}}\left(q^{n}-\sum _{\ell \mid n,\ \ell {\text{ prime}}}q^{n/\ell }\right);}
7401:
3474:
2095:
2137:
which make the needed
Euclidean divisions very efficient. However, for some fields, typically in characteristic
781:
8460:
4694:
3962:
2366:
1359:
1321:
7721:. More precisely, this polynomial is the product of all monic polynomials of degree one over a field of order
1988:. The multiplicative inverse of a non-zero element may be computed with the extended Euclidean algorithm; see
1588:
9841:
9381:
7909:
7685:
402:
9021:
8324:, since computer data is stored in binary. For example, a byte of data can be interpreted as an element of
9309:
9098:
7899:
This property is used to compute the product of the irreducible factors of each degree of polynomials over
9166:
8688:
9836:
7660:
6573:
6183:
5762:
271:
9533:
Shparlinski, Igor E. (2013), "Additive
Combinatorics over Finite Fields: New Results and Applications",
8068:
9813:
8644:
8278:
5901:
5799:
2247:
1115:
9234:
658:
9643:
9482:
8476:
8372:
7291:
8930:
The formal validation of this notation results from the fact that the above field inclusions form a
8439:
8371:
proceed by reduction modulo one or several primes, and then reconstruction of the solution by using
7674:
has functions for factoring polynomials over finite fields, or, at least, over finite prime fields.
9419:
8435:
8360:
4743:
4031:
613:
40:
9831:
8724:
9404:
7671:
2086:, which produce isomorphic results. To simplify the Euclidean division, one commonly chooses for
362:
9063:
7916:
9871:
9399:
9372:
9364:
8480:
7637:
7287:
5905:
5530:
2356:
1815:
1008:
57:
8074:
By the above formula, the number of irreducible (not necessarily monic) polynomials of degree
8830:
8336:. Some CPUs have special instructions that can be useful for finite fields of characteristic
8309:
8251:, the right hand side is positive, so there is at least one irreducible polynomial of degree
6815:
6608:
6313:
6275:
3335:
450:
8991:
5647:
9282:
9202:
9139:
8400:
is an example of a deep result involving many mathematical tools, including finite fields.
7304:
1503:
The uniqueness up to isomorphism of splitting fields implies thus that all fields of order
1302:
1090:
736:
732:
349:
341:
313:
308:
299:
256:
198:
9689:
4158:
may thus be defined as follows; in following formulas, the operations between elements of
8:
9801:
9450:(1896), "A doubly-infinite system of simple groups", in E. H. Moore; et al. (eds.),
5973:
5965:
allows one to solve this problem by constructing the table of the discrete logarithms of
3415:
as a quadratic non-residue, which allows us to have a very simple irreducible polynomial
3161:
has to remain undefined.) From the tables, it can be seen that the additive structure of
1457:
1264:
1076:
549:
446:
367:
357:
208:
108:
100:
91:
9806:
7562:{\displaystyle \mathrm {Id} =\varphi ^{0},\varphi ,\varphi ^{2},\ldots ,\varphi ^{n-1}.}
2240:
The smallest non-prime field is the field with four elements, which is commonly denoted
747:
of the field. (In general there will be several primitive elements for a given field.)
9598:
9570:
9523:
This can be verified by looking at the information on the page provided by the browser.
9468:
in an address given in 1893 at the
International Mathematical Congress held in Chicago
9394:
8431:
8393:
8352:
8341:
8286:
7646:, if it is not the product of two non-constant monic polynomials, with coefficients in
7267:
6818:
to each other, a root and its (multiplicative) inverse do not belong to the same orbit.
5910:
5876:
3154:
831:
770:
521:
173:
164:
122:
8923:{\displaystyle {\overline {\mathbb {F} }}_{p}=\bigcup _{n\geq 1}\mathbb {F} _{p^{n}}.}
8054:
442:
9817:
9787:
9769:
9751:
9675:
9648:
9588:
9546:
8467:
are commutative, and hence are finite fields. This result holds even if we relax the
8408:
7135:
They split into six orbits of six elements each under the action of the Galois group.
6374:) are primitive elements; and the primitive elements are not all conjugate under the
4906:
may be defined as follows; in following formulas, the operations between elements of
4890:
is defined as a root of the given irreducible polynomial). As the characteristic of
1422:
454:
9737:
9723:
9602:
9733:
9719:
9685:
9638:
9628:
9580:
9538:
9376:
8404:
7667:
7627:
7584:
6500:
3166:
2342:
1086:
743:, so all non-zero elements can be expressed as powers of a single element called a
513:
193:
20:
6814:
They form two orbits under the action of the Galois group. As the two factors are
218:
9743:
9671:
9584:
8412:
8388:
8368:
8298:
8282:
7795:
7656:
7602:
The fact that the
Frobenius map is surjective implies that every finite field is
2062:(for example after a multiplication), one knows that one has to use the relation
1556:
In summary, we have the following classification theorem first proved in 1893 by
1353:
1260:
652:
529:
483:
of a finite field is its number of elements, which is either a prime number or a
285:
279:
266:
246:
237:
203:
140:
9855:
8376:
8364:
5995:
750:
The simplest examples of finite fields are the fields of prime order: for each
458:
327:
9633:
9616:
9542:
9865:
9652:
9424:
8472:
8468:
8464:
8456:
8423:
8416:
8380:
8348:
8294:
7917:
Number of monic irreducible polynomials of a given degree over a finite field
7603:
7572:
6008:
5654:, that is, all non-zero elements are powers of a single element. In summary:
5572:
3158:
1838:
1313:
1306:
537:
525:
517:
213:
178:
135:
8820:{\displaystyle \mathbb {\mathbb {F} } _{p^{n}}\subset \mathbb {F} _{p^{nm}}}
2406:
the construction of the preceding section must involve this polynomial, and
9414:
8935:
8931:
8356:
8302:
8274:
7596:
6375:
6292:
5744:, the primitive element is not unique. The number of primitive elements is
5651:
983:
751:
740:
553:
533:
473:
387:
318:
152:
8396:
were motivated by the need to enlarge the power of these modular methods.
3169:, while the non-zero multiplicative structure is isomorphic to the group Z
2348:
This may be deduced as follows from the results of the preceding section.
9465:
9447:
9409:
8427:
6324:
into distinct irreducible polynomials that have all the same degree, say
1814:
may be explicitly constructed in the following way. One first chooses an
1557:
1343:
760:
580:
506:
484:
430:
377:
372:
261:
251:
225:
2214:, with an even number of terms, are never irreducible in characteristic
1324:, any finite division ring is commutative, and hence is a finite field.
9368:
8438:
finite fields and finite field models are used extensively, such as in
7263:
713:
642:
127:
3344:(this is almost the definition of a quadratic non-residue). There are
3283:
is an odd prime, there are always irreducible polynomials of the form
512:
Finite fields are fundamental in a number of areas of mathematics and
9575:
2188:
1991:
Extended
Euclidean algorithm § Simple algebraic field extensions
1582:
and they are all isomorphic. In these fields, every element satisfies
382:
188:
145:
113:
9565:
Green, Ben (2005), "Finite field models in additive combinatorics",
8351:, as many problems over the integers may be solved by reducing them
5987:(it is convenient to define the discrete logarithm of zero as being
8317:
8040:{\displaystyle N(q,n)={\frac {1}{n}}\sum _{d\mid n}\mu (d)q^{n/d},}
6807:{\displaystyle (X^{6}+X^{4}+X^{2}+X+1)(X^{6}+X^{5}+X^{4}+X^{2}+1).}
5932:. However, addition amounts to computing the discrete logarithm of
1565:
The order of a finite field is a prime power. For every prime power
183:
9728:
W. H. Bussey (1910) "Tables of Galois fields of order < 1000",
8411:
over finite fields and the theory has many applications including
6422:
6259:
1962:. The addition and the subtraction are those of polynomials over
1738:; in that case, this subfield is unique. In fact, the polynomial
868:
9617:"Finite field models in arithmetic combinatorics – ten years on"
8685:
This property results mainly from the fact that the elements of
19:"Galois field" redirects here. For Galois field extensions, see
8329:
2073:
to reduce its degree (it is what
Euclidean division is doing).
117:
9359:
Although finite fields are not algebraically closed, they are
8980:
8057:. This formula is an immediate consequence of the property of
2498:{\displaystyle \mathrm {GF} (4)=\mathrm {GF} (2)/(X^{2}+X+1).}
651:
below). Moreover, a field cannot contain two different finite
8674:
8290:
5994:
Zech's logarithms are useful for large computations, such as
3270:, one has to find an irreducible polynomial of degree 2. For
3216:
7677:
2341:, the other operation results being easily deduced from the
8285:
is the basis of several widely used protocols, such as the
7158:
6672:
and are all conjugate under the action of the Galois group.
6243:
th roots of unity never exist in a field of characteristic
3531:
are defined as follows (the operations between elements of
1222:{\displaystyle X^{p}-X=\prod _{a\in \mathrm {GF} (p)}(X-a)}
612:
copies of any element always results in zero; that is, the
6600:
rd roots of unity. Summing these numbers, one finds again
4014:(to show this, it suffices to show that it has no root in
9668:
Points and lines. Characterizing the classical geometries
7750:
is the product of all monic irreducible polynomials over
5518:
as integer powers). These elements are the four roots of
1837:(such an irreducible polynomial always exists). Then the
1456:, which in general implies that the splitting field is a
8562:{\displaystyle f(T)=1+\prod _{\alpha \in F}(T-\alpha ),}
8320:. The finite field almost always has characteristic of
5539:
is a primitive element, and the primitive elements are
559:
The number of elements of a finite field is called its
4166:, represented by Latin letters, are the operations in
1970:. The product of two elements is the remainder of the
9312:
9285:
9237:
9205:
9169:
9142:
9101:
9066:
9024:
8994:
8944:
8859:
8833:
8770:
8727:
8691:
8647:
8611:
8503:
8117:
7956:
7688:
7490:
7404:
7325:
7221:
6835:
6685:
6629:
4920:
4910:, represented by Latin letters are the operations in
4851:
4802:{\displaystyle a+b\alpha +c\alpha ^{2}+d\alpha ^{3},}
4751:
4697:
4180:
4150:
The addition, additive inverse and multiplication on
4115:
4039:
3965:
3553:
3477:
3182:
2412:
2369:
2250:
2098:
1910:{\displaystyle \mathrm {GF} (q)=\mathrm {GF} (p)/(P)}
1846:
1643:
1591:
1476:, as well as the multiplicative inverse of a root of
1362:
1154:
1016:
784:
661:
7398:is not the identity, as, otherwise, the polynomial
5658:
The multiplicative group of the non-zero elements in
2225:
A possible choice for such a polynomial is given by
2187:
that makes the polynomial irreducible. If all these
917:
of the field. This allows defining a multiplication
843:
Extended
Euclidean algorithm § Modular integers
3539:represented by Latin letters are the operations in
3135:must be read in the left column, and the values of
1464:shows that the sum and the product of two roots of
9805:
9337:
9298:
9271:
9223:
9191:
9155:
9126:
9087:
9052:
9010:
8977:which may thus be considered as "directed union".
8969:
8922:
8845:
8819:
8756:
8713:
8665:
8633:
8561:
8426:, two well known examples being the definition of
8225:
8039:
7715:factors into linear factors over a field of order
7707:
7561:
7430:
7367:
7246:
7127:
6806:
6664:
6544:. As the 3rd and the 7th roots of unity belong to
5496:
4876:
4801:
4722:
4676:
4140:
4073:
3990:
3944:
3495:
3277:, this has been done in the preceding section. If
3215:is the non-trivial field automorphism, called the
3207:
2497:
2394:
2268:
2129:
1909:
1699:
1613:
1387:
1221:
1105:, except the first and the last, is a multiple of
1067:
830:. The sum, the difference and the product are the
805:
676:
735:. The non-zero elements of a finite field form a
703:, where the letters GF stand for "Galois field".
9863:
9799:
8359:. For example, the fastest known algorithms for
7375:It has been shown in the preceding section that
7368:{\displaystyle \varphi ^{k}:x\mapsto x^{p^{k}}.}
5900:can be computed very quickly, for example using
5514:(the elements that have all nonzero elements of
8970:{\displaystyle {\overline {\mathbb {F} }}_{p},}
7616:Factorization of polynomials over finite fields
2345:. See below for the complete operation tables.
1997:However, with this representation, elements of
9492:National Institute of Standards and Technology
9484:Recommended Elliptic Curves for Government Use
8634:{\displaystyle {\overline {\mathbb {F} }}_{p}}
8450:
6278:are distinct in every field of characteristic
3232:of the above mentioned irreducible polynomial
1305:. To use a piece of jargon, finite fields are
9781:
9763:
9730:Bulletin of the American Mathematical Society
9716:Bulletin of the American Mathematical Society
9707:W. H. Bussey (1905) "Galois field tables for
9569:, Cambridge University Press, pp. 1–28,
9469:
8422:Finite fields have widespread application in
7139:This shows that the best choice to construct
6007:Every nonzero element of a finite field is a
1700:{\displaystyle X^{q}-X=\prod _{a\in F}(X-a).}
647:
410:
9306:is the primitive element already chosen for
8497:is not algebraically closed: the polynomial
7278:, which fixes every element of the subfield
6235:th root of unity. It follows that primitive
3246:
1409:is a finite field of lowest order, in which
986:. It follows that the number of elements of
822:may be represented by integers in the range
600:is a positive integer). In a field of order
9532:
8981:Primitive elements in the algebraic closure
7936:of monic irreducible polynomials of degree
7609:
5558:
2235:
2050:, and, when one encounters a polynomial in
1327:
9354:
8853:These inclusions allow writing informally
6131:th primitive root of unity if and only if
5650:implies that this multiplicative group is
5648:structure theorem of finite abelian groups
5361:
5349:
4898:, each element is its additive inverse in
3453:, in the same way that the complex number
3443:, that is, a symbol that has the property
2191:are reducible, one chooses "pentanomials"
2170:is reducible, it is recommended to choose
1500:by the minimality of the splitting field.
962:by choosing an integer representative for
806:{\displaystyle \mathbb {Z} /p\mathbb {Z} }
453:. As with any field, a finite field is a
417:
403:
9782:Mullen, Gary L.; Panario, Daniel (2013),
9644:1983/d340f853-0584-49c8-a463-ea16ee51ce0f
9642:
9632:
9574:
9315:
9172:
9104:
9027:
8949:
8900:
8864:
8797:
8774:
8694:
8650:
8616:
8111:The exact formula implies the inequality
7832:, it defines a field extension of degree
7678:Irreducible polynomials of a given degree
5668:is cyclic, i.e., there exists an element
2253:
2080:, there are several possible choices for
1547:cannot contain another subfield of order
816:The elements of the prime field of order
799:
786:
664:
548:A finite field is a finite set that is a
76:Learn how and when to remove this message
9742:
9511:
7812:is an irreducible monic polynomial over
7159:Frobenius automorphism and Galois theory
6338:is the smallest field of characteristic
4074:{\displaystyle a+b\alpha +c\alpha ^{2},}
2210:, as polynomials of degree greater than
1773:
39:This article includes a list of general
9764:Mullen, Gary L.; Mummert, Carl (2007),
9750:(Second ed.), Dover Publications,
9464:This latter notation was introduced by
9454:, Macmillan & Co., pp. 208–242
9442:
9440:
9347:Such a construction may be obtained by
7247:{\displaystyle \varphi :x\mapsto x^{p}}
6052:that is not a solution of the equation
3208:{\displaystyle \varphi :x\mapsto x^{2}}
1079:) is true in a field of characteristic
9864:
9829:
9768:, Student Mathematical Library (AMS),
9053:{\displaystyle \mathbb {F} _{q^{mn}},}
6081:th primitive root of unity in a field
5555:(that is, 1, 2, 4, 7, 8, 11, 13, 14).
4902:. The addition and multiplication on
4141:{\displaystyle \alpha ^{3}=\alpha +1.}
3425:Having chosen a quadratic non-residue
2557:result from this, and are as follows:
2141:, irreducible polynomials of the form
9665:
9564:
9446:
9338:{\displaystyle \mathbb {F} _{q^{m}}.}
9127:{\displaystyle \mathbb {F} _{q^{n}}.}
8398:Wiles' proof of Fermat's Last Theorem
6284:, as this polynomial is a divisor of
5805:
4877:{\displaystyle \alpha ^{4}=\alpha +1}
3259:
1068:{\displaystyle (x+y)^{p}=x^{p}+y^{p}}
9621:Finite Fields and Their Applications
9614:
9535:Finite Fields and Their Applications
9437:
9192:{\displaystyle \mathbb {F} _{q^{n}}}
8714:{\displaystyle \mathbb {F} _{p^{n}}}
8486:
8239:is a power of some prime. For every
6507:. This may be verified by factoring
4734:, that is, it is irreducible modulo
4006:, that is, it is irreducible modulo
2513:denote a root of this polynomial in
1525:as a subfield, its elements are the
1297:by adjoining a single element whose
1237:. More generally, every element in
25:
5575:under the multiplication, of order
4022:). It follows that the elements of
3954:
2019:that corresponds to the polynomial
1956:whose degree is strictly less than
1778:
855:be a finite field. For any element
13:
8314:Reed–Solomon error correction code
7670:. At least for this reason, every
7626:is a finite field, a non-constant
7495:
7492:
6353:
4738:. It follows that the elements of
4288:
4285:
2553:. The tables of the operations in
2437:
2434:
2417:
2414:
1871:
1868:
1851:
1848:
1718:contains a subfield isomorphic to
1245:satisfies the polynomial equation
1189:
1186:
45:it lacks sufficient corresponding
14:
9883:
9849:
8666:{\displaystyle \mathbb {F} _{p}.}
8430:and the related construction for
8347:Finite fields are widely used in
6002:
5640:is the lowest possible value for
2276:It consists of the four elements
2269:{\displaystyle \mathbb {F} _{4}.}
2155:may not exist. In characteristic
1940:More explicitly, the elements of
1509:are isomorphic. Also, if a field
449:that contains a finite number of
9766:Finite Fields and Applications I
9537:, DE GRUYTER, pp. 233–272,
9272:{\displaystyle g_{m}=g_{n}^{h},}
8407:concern the number of points on
8297:, many codes are constructed as
8293:) over a large finite field. In
5824:, then for any non-zero element
5563:The set of non-zero elements in
3262:of finite fields in the case of
731:elements of the finite field as
677:{\displaystyle \mathbb {F} _{q}}
30:
9738:10.1090/S0002-9904-1910-01888-7
9724:10.1090/S0002-9904-1905-01284-2
9501:from the original on 2008-07-19
8308:Finite fields are used by many
8268:
6679:st roots of unity are roots of
6623:th roots of unity are roots of
6157:, then the number of primitive
5360:
4306:this operation is the identity)
4278:
3471:are all the linear expressions
3386:is a quadratic non-residue for
3375:is a quadratic non-residue for
3310:More precisely, the polynomial
1482:. In other words, the roots of
648:§ Existence and uniqueness
9659:
9608:
9558:
9526:
9517:
9505:
9475:
9458:
8764:and this defines an inclusion
8553:
8541:
8513:
8507:
8392:. Many recent developments of
8133:
8121:
8013:
8007:
7972:
7960:
7770:is an irreducible factor over
7342:
7231:
7119:
7068:
7065:
7014:
7011:
6953:
6950:
6918:
6915:
6890:
6887:
6836:
6798:
6740:
6737:
6686:
6665:{\displaystyle X^{6}+X^{3}+1,}
6381:The order of this field being
6045:is a solution of the equation
5768:The result above implies that
5477:
5432:
5416:
5362:
5343:
5280:
5274:
5238:
5228:
5181:
5178:
5131:
5114:
5102:
5086:
5074:
5065:
5053:
5047:
5035:
5025:
4978:
4972:
4925:
4657:
4621:
4612:
4567:
4561:
4534:
4524:
4493:
4490:
4459:
4442:
4430:
4421:
4409:
4403:
4391:
4381:
4350:
4344:
4313:
4298:
4292:
4265:
4256:
4247:
4238:
4219:
4188:
3923:
3893:
3890:
3881:
3866:
3836:
3814:
3798:
3788:
3770:
3764:
3743:
3733:
3718:
3715:
3700:
3690:
3678:
3672:
3660:
3650:
3635:
3629:
3614:
3604:
3595:
3576:
3561:
3365:quadratic non-residues modulo
3192:
2489:
2464:
2456:
2450:
2447:
2441:
2427:
2421:
2076:Except in the construction of
2056:of degree greater or equal to
1904:
1898:
1890:
1884:
1881:
1875:
1861:
1855:
1691:
1679:
1216:
1204:
1199:
1193:
1030:
1017:
1:
9700:
9567:Surveys in Combinatorics 2005
8445:
8233:this is sharp if and only if
7910:Distinct degree factorization
6350:th primitive roots of unity.
6215:In a field of characteristic
6018:for every nonzero element of
5917:When the nonzero elements of
3459:is a symbolic square root of
3437:be a symbolic square root of
1461:
1148:. This implies the equality
1011:
543:
9585:10.1017/cbo9780511734885.002
8953:
8868:
8757:{\displaystyle x^{p^{n}}-x,}
8620:
8442:on arithmetic progressions.
5836:, there is a unique integer
5792:. The particular case where
968:. This multiplication makes
7:
9837:Encyclopedia of Mathematics
9388:
8641:be an algebraic closure of
8461:Wedderburn's little theorem
8451:Wedderburn's little theorem
7661:unique factorization domain
7484:-automorphisms, which are
7431:{\displaystyle X^{p^{k}}-X}
6562:th roots of unity for some
6534:th roots of unity for some
6229:th root of unity is also a
5551:less than and coprime with
3496:{\displaystyle a+b\alpha ,}
2130:{\displaystyle X^{n}+aX+b,}
1322:Wedderburn's little theorem
706:In a finite field of order
493:and every positive integer
10:
9888:
9814:Cambridge University Press
9361:quasi-algebraically closed
9095:is a primitive element of
9088:{\displaystyle g_{mn}^{m}}
8479:are finite fields, by the
8477:alternative division rings
8340:, generally variations of
8279:discrete logarithm problem
7613:
6476:. This implies that, over
6099:roots of unity, which are
6035:is a positive integer, an
5902:exponentiation by squaring
5816:is a primitive element in
3260:above general construction
2092:a polynomial of the form
1925:by the ideal generated by
769:may be constructed as the
567:. A finite field of order
499:there are fields of order
18:
9784:Handbook of Finite Fields
9666:Shult, Ernest E. (2011).
9634:10.1016/j.ffa.2014.11.003
9543:10.1515/9783110283600.233
9470:Mullen & Panario 2013
9382:Chevalley–Warning theorem
9363:, which means that every
8721:are exactly the roots of
8373:Chinese remainder theorem
7292:Ferdinand Georg Frobenius
6556:generators are primitive
6059:for any positive integer
5586:, there exists a divisor
4723:{\displaystyle X^{4}+X+1}
4686:
3991:{\displaystyle X^{3}-X-1}
3141:in the top row. (Because
2961:
2784:
2602:
2588:
2576:
2564:
2395:{\displaystyle X^{2}+X+1}
2181:with the lowest possible
1948:are the polynomials over
1573:there are fields of order
1388:{\displaystyle P=X^{q}-X}
1267:and simple. That is, if
1085:. This follows from the
832:remainder of the division
487:. For every prime number
9670:. Universitext. Berlin:
9494:, July 1999, p. 3,
9430:
9420:Elementary abelian group
9199:in order that, whenever
8436:arithmetic combinatorics
8361:polynomial factorization
8277:, the difficulty of the
8069:Möbius inversion formula
7838:, which is contained in
7610:Polynomial factorization
6574:Euler's totient function
6456:elements. The remaining
6184:Euler's totient function
5763:Euler's totient function
5633:solutions in any field,
5559:Multiplicative structure
3463:. Then, the elements of
2236:Field with four elements
1614:{\displaystyle x^{q}=x,}
1328:Existence and uniqueness
9830:Skopin, A. I. (2001) ,
9615:Wolf, J. (March 2015).
9405:Finite field arithmetic
9355:Quasi-algebraic closure
8846:{\displaystyle m>1.}
8463:states that all finite
8281:in finite fields or in
7758:, whose degree divides
7708:{\displaystyle X^{q}-X}
7672:computer algebra system
7169:is a prime number, and
6592:st roots of unity, and
6043:primitive root of unity
5906:cryptographic protocols
5800:Fermat's little theorem
5531:multiplicative inverses
4845:is a symbol such that
4109:is a symbol such that
1917:of the polynomial ring
1116:Fermat's little theorem
60:more precise citations.
9400:Field with one element
9365:homogeneous polynomial
9339:
9300:
9273:
9225:
9193:
9157:
9128:
9089:
9054:
9012:
9011:{\displaystyle g_{mn}}
8971:
8924:
8847:
8821:
8758:
8715:
8673:It is not only unique
8667:
8635:
8563:
8475:, that is, all finite
8310:error correction codes
8227:
8041:
7730:This implies that, if
7709:
7563:
7432:
7369:
7288:Frobenius automorphism
7248:
7129:
6825:primitive elements of
6808:
6666:
6609:cyclotomic polynomials
6425:, the intersection of
6385:, and the divisors of
6306:. It follows that the
6252:On the other hand, if
5498:
4878:
4803:
4742:may be represented by
4724:
4678:
4142:
4075:
4030:may be represented by
3992:
3946:
3497:
3217:Frobenius automorphism
3209:
2499:
2396:
2357:irreducible polynomial
2270:
2131:
2033:become polynomials in
2025:. So, the elements of
1911:
1816:irreducible polynomial
1708:
1701:
1615:
1488:form a field of order
1460:of the original). The
1389:
1273:is a finite field and
1223:
1124:is a prime number and
1075:(sometimes called the
1069:
911:is the characteristic
807:
678:
635:, all fields of order
594:is a prime number and
573:exists if and only if
441:(so-named in honor of
9340:
9301:
9299:{\displaystyle g_{m}}
9274:
9226:
9224:{\displaystyle n=mh,}
9194:
9158:
9156:{\displaystyle g_{n}}
9129:
9090:
9055:
9013:
8972:
8925:
8848:
8822:
8759:
8716:
8668:
8636:
8601:Given a prime number
8564:
8228:
8042:
7788:, its degree divides
7710:
7630:with coefficients in
7564:
7438:would have more than
7433:
7381:is the identity. For
7370:
7249:
7215:implies that the map
7130:
6809:
6675:The twelve primitive
6667:
6576:shows that there are
6314:cyclotomic polynomial
6276:cyclotomic polynomial
6192:th roots of unity in
6163:th roots of unity in
5499:
4879:
4804:
4725:
4679:
4143:
4076:
3993:
3947:
3498:
3336:quadratic non-residue
3225:into the second root
3210:
3165:is isomorphic to the
2500:
2397:
2271:
2132:
1912:
1774:Explicit construction
1702:
1616:
1562:
1515:has a field of order
1395:over the prime field
1390:
1263:of a finite field is
1229:for polynomials over
1224:
1070:
898:. The least positive
808:
679:
9858:at Wolfram research.
9802:Niederreiter, Harald
9310:
9283:
9235:
9203:
9167:
9140:
9099:
9064:
9022:
8992:
8942:
8857:
8831:
8768:
8725:
8689:
8645:
8609:
8501:
8305:over finite fields.
8115:
7954:
7686:
7488:
7402:
7323:
7219:
6833:
6683:
6627:
6552:, respectively, the
6480:, there are exactly
5686:non-zero elements of
4918:
4849:
4749:
4730:is irreducible over
4695:
4178:
4113:
4103:(respectively), and
4037:
3998:is irreducible over
3963:
3551:
3523:. The operations on
3475:
3320:is irreducible over
3180:
2545:are the elements of
2517:. This implies that
2410:
2367:
2355:, there is only one
2248:
2159:, if the polynomial
2096:
1931:is a field of order
1844:
1783:Given a prime power
1641:
1589:
1494:, which is equal to
1421:distinct roots (the
1360:
1152:
1093:of the expansion of
1091:binomial coefficient
1014:
782:
737:multiplicative group
659:
314:Group with operators
257:Complemented lattice
92:Algebraic structures
9265:
9084:
8440:Szemerédi's theorem
8409:algebraic varieties
8332:bar code, which is
8328:. One exception is
7860:, and are roots of
7846:, and all roots of
7447:There are no other
7392:, the automorphism
7286:. It is called the
7264:linear endomorphism
7143:is to define it as
6584:th roots of unity,
6437:is the prime field
6393:, the subfields of
6268:, the roots of the
5606:for every non-zero
3405:= 3, 7, 11, 19, ...
3380:= 3, 5, 11, 13, ...
3251:) for an odd prime
2561:
1458:separable extension
1356:of the polynomial
563:or, sometimes, its
505:, all of which are
368:Composition algebra
128:Quasigroup and loop
16:Algebraic structure
9395:Quasi-finite field
9349:Conway polynomials
9335:
9296:
9269:
9251:
9221:
9189:
9153:
9124:
9085:
9067:
9050:
9008:
8967:
8920:
8897:
8843:
8817:
8754:
8711:
8663:
8631:
8559:
8540:
8481:Artin–Zorn theorem
8394:algebraic geometry
8367:over the field of
8342:carry-less product
8223:
8196:
8037:
8003:
7705:
7659:over a field is a
7575:, this means that
7559:
7463:. In other words,
7455:-automorphisms of
7428:
7365:
7268:field automorphism
7244:
7125:
6829:are the roots of
6804:
6662:
6619:The six primitive
6615:, one finds that:
6364:minimal polynomial
6344:that contains the
6300:is nonzero modulo
5911:Discrete logarithm
5877:discrete logarithm
5806:Discrete logarithm
5620:. As the equation
5584:Lagrange's theorem
5512:primitive elements
5494:
5492:
4874:
4799:
4720:
4674:
4672:
4138:
4071:
3988:
3942:
3940:
3493:
3205:
2560:
2495:
2392:
2266:
2227:Conway polynomials
2127:
2011:to the element of
1980:of the product in
1972:Euclidean division
1907:
1697:
1678:
1622:and the polynomial
1611:
1403:. This means that
1385:
1299:minimal polynomial
1219:
1203:
1065:
803:
674:
522:algebraic geometry
9793:978-1-4398-7378-6
9775:978-0-8218-4418-2
9757:978-0-486-47189-1
9681:978-3-642-15626-7
8987:primitive element
8956:
8882:
8871:
8623:
8525:
8487:Algebraic closure
8432:Hadamard Matrices
8193:
8186:
8167:
8147:
7988:
7986:
7806:. Conversely, if
7163:In this section,
6607:By factoring the
6501:monic polynomials
6186:). The number of
6093:contains all the
5974:Zech's logarithms
5727:primitive element
5533:. In particular,
4307:
4282:
3407:, one may choose
3258:For applying the
3115:
3114:
3111:
3110:
2959:
2958:
2782:
2781:
1663:
1423:formal derivative
1291:is obtained from
1279:is a subfield of
1174:
998:for some integer
745:primitive element
427:
426:
86:
85:
78:
9879:
9844:
9826:
9812:(2nd ed.),
9811:
9796:
9778:
9760:
9744:Jacobson, Nathan
9732:16(4): 188–206,
9713:
9694:
9693:
9663:
9657:
9656:
9646:
9636:
9612:
9606:
9605:
9578:
9562:
9556:
9555:
9530:
9524:
9521:
9515:
9509:
9503:
9502:
9500:
9489:
9479:
9473:
9462:
9456:
9455:
9444:
9344:
9342:
9341:
9336:
9331:
9330:
9329:
9328:
9318:
9305:
9303:
9302:
9297:
9295:
9294:
9278:
9276:
9275:
9270:
9264:
9259:
9247:
9246:
9230:
9228:
9227:
9222:
9198:
9196:
9195:
9190:
9188:
9187:
9186:
9185:
9175:
9162:
9160:
9159:
9154:
9152:
9151:
9133:
9131:
9130:
9125:
9120:
9119:
9118:
9117:
9107:
9094:
9092:
9091:
9086:
9083:
9078:
9059:
9057:
9056:
9051:
9046:
9045:
9044:
9043:
9030:
9017:
9015:
9014:
9009:
9007:
9006:
8976:
8974:
8973:
8968:
8963:
8962:
8957:
8952:
8947:
8929:
8927:
8926:
8921:
8916:
8915:
8914:
8913:
8903:
8896:
8878:
8877:
8872:
8867:
8862:
8852:
8850:
8849:
8844:
8826:
8824:
8823:
8818:
8816:
8815:
8814:
8813:
8800:
8791:
8790:
8789:
8788:
8778:
8777:
8763:
8761:
8760:
8755:
8744:
8743:
8742:
8741:
8720:
8718:
8717:
8712:
8710:
8709:
8708:
8707:
8697:
8681:
8672:
8670:
8669:
8664:
8659:
8658:
8653:
8640:
8638:
8637:
8632:
8630:
8629:
8624:
8619:
8614:
8604:
8597:
8591:
8585:
8574:
8569:has no roots in
8568:
8566:
8565:
8560:
8539:
8496:
8405:Weil conjectures
8369:rational numbers
8339:
8335:
8327:
8323:
8264:
8256:
8250:
8244:
8238:
8232:
8230:
8229:
8224:
8219:
8215:
8214:
8213:
8209:
8195:
8194:
8191:
8184:
8163:
8162:
8148:
8140:
8107:
8087:
8079:
8066:
8052:
8046:
8044:
8043:
8038:
8033:
8032:
8028:
8002:
7987:
7979:
7949:
7941:
7935:
7906:
7895:
7885:
7875:
7869:
7859:
7851:
7845:
7837:
7831:
7825:
7819:
7811:
7805:
7798:is contained in
7793:
7787:
7777:
7769:
7763:
7757:
7749:
7739:
7726:
7720:
7714:
7712:
7711:
7706:
7698:
7697:
7668:rational numbers
7651:
7645:
7635:
7628:monic polynomial
7625:
7594:
7585:Galois extension
7582:
7568:
7566:
7565:
7560:
7555:
7554:
7530:
7529:
7511:
7510:
7498:
7483:
7476:
7470:
7462:
7454:
7443:
7437:
7435:
7434:
7429:
7421:
7420:
7419:
7418:
7397:
7391:
7380:
7374:
7372:
7371:
7366:
7361:
7360:
7359:
7358:
7335:
7334:
7319:times, we have
7318:
7312:
7302:
7285:
7277:
7261:
7253:
7251:
7250:
7245:
7243:
7242:
7214:
7195:
7184:
7178:
7168:
7154:
7142:
7134:
7132:
7131:
7126:
7106:
7105:
7093:
7092:
7080:
7079:
7052:
7051:
7039:
7038:
7026:
7025:
7004:
7003:
6991:
6990:
6978:
6977:
6965:
6964:
6943:
6942:
6930:
6929:
6902:
6901:
6874:
6873:
6861:
6860:
6848:
6847:
6828:
6824:
6813:
6811:
6810:
6805:
6791:
6790:
6778:
6777:
6765:
6764:
6752:
6751:
6724:
6723:
6711:
6710:
6698:
6697:
6678:
6671:
6669:
6668:
6663:
6652:
6651:
6639:
6638:
6622:
6614:
6603:
6599:
6595:
6591:
6587:
6583:
6579:
6571:
6567:
6561:
6555:
6551:
6547:
6543:
6539:
6533:
6527:
6524:The elements of
6520:
6516:
6506:
6498:
6497:
6495:
6494:
6491:
6488:
6479:
6475:
6471:
6467:
6463:
6459:
6455:
6451:
6447:
6440:
6436:
6432:
6428:
6420:
6416:
6412:
6408:
6404:
6400:
6396:
6392:
6388:
6384:
6373:
6369:
6361:
6349:
6343:
6337:
6329:
6323:
6311:
6305:
6299:
6290:
6283:
6273:
6267:
6257:
6248:
6242:
6234:
6228:
6220:
6211:
6199:
6191:
6181:
6170:
6162:
6156:
6150:is a divisor of
6149:
6143:
6137:is a divisor of
6136:
6130:
6124:
6113:
6098:
6092:
6086:
6080:
6074:
6068:
6058:
6051:
6040:
6034:
6025:
6017:
5990:
5986:
5971:
5962:
5942:. The identity
5941:
5931:
5924:
5899:
5890:
5884:
5874:
5865:
5853:
5841:
5835:
5829:
5823:
5815:
5797:
5791:
5783:
5777:
5760:
5754:
5743:
5736:
5724:
5719:Such an element
5715:
5694:
5684:
5674:
5666:
5645:
5639:
5632:
5626:
5619:
5611:
5605:
5598:
5591:
5581:
5570:
5554:
5550:
5544:
5538:
5528:
5517:
5509:
5503:
5501:
5500:
5495:
5493:
5489:
5488:
5428:
5427:
5356:
5227:
5226:
5211:
5210:
5177:
5176:
5161:
5160:
5126:
5125:
5098:
5097:
5024:
5023:
5008:
5007:
4971:
4970:
4955:
4954:
4913:
4909:
4905:
4901:
4897:
4893:
4889:
4883:
4881:
4880:
4875:
4861:
4860:
4844:
4838:
4834:
4830:
4826:
4808:
4806:
4805:
4800:
4795:
4794:
4779:
4778:
4741:
4737:
4733:
4729:
4727:
4726:
4721:
4707:
4706:
4683:
4681:
4680:
4675:
4673:
4669:
4668:
4523:
4522:
4489:
4488:
4454:
4453:
4380:
4379:
4343:
4342:
4308:
4305:
4291:
4283:
4280:
4277:
4276:
4218:
4217:
4174:, respectively:
4173:
4169:
4165:
4161:
4157:
4153:
4147:
4145:
4144:
4139:
4125:
4124:
4108:
4102:
4098:
4095:are elements of
4094:
4080:
4078:
4077:
4072:
4067:
4066:
4029:
4025:
4021:
4017:
4013:
4009:
4005:
4001:
3997:
3995:
3994:
3989:
3975:
3974:
3955:GF(8) and GF(27)
3951:
3949:
3948:
3943:
3941:
3934:
3933:
3921:
3920:
3905:
3904:
3877:
3876:
3864:
3863:
3848:
3847:
3825:
3824:
3546:
3538:
3530:
3522:
3514:
3508:
3502:
3500:
3499:
3494:
3470:
3462:
3458:
3452:
3442:
3436:
3430:
3421:
3414:
3406:
3399:
3392:
3385:
3381:
3374:
3370:
3364:
3363:
3361:
3360:
3357:
3354:
3343:
3333:
3327:
3319:
3306:
3298:
3292:
3282:
3276:
3269:
3242:
3231:
3224:
3214:
3212:
3211:
3206:
3204:
3203:
3167:Klein four-group
3164:
3152:
3148:
3140:
3134:
3129:, the values of
3128:
3122:
3107:
3102:
3095:
3087:
3077:
3069:
3064:
3057:
3048:
3041:
3033:
3028:
3021:
3016:
3011:
3006:
2999:
2991:
2984:
2978:
2971:
2963:
2962:
2955:
2948:
2943:
2935:
2930:
2920:
2915:
2907:
2900:
2895:
2886:
2878:
2871:
2866:
2861:
2854:
2849:
2844:
2839:
2834:
2827:
2819:
2812:
2807:
2801:
2794:
2786:
2785:
2778:
2773:
2768:
2761:
2753:
2743:
2738:
2733:
2725:
2718:
2709:
2702:
2694:
2689:
2684:
2677:
2669:
2662:
2657:
2652:
2645:
2637:
2630:
2625:
2619:
2612:
2604:
2603:
2598:
2586:
2574:
2562:
2559:
2556:
2552:
2549:that are not in
2548:
2544:
2537:
2528:
2516:
2512:
2504:
2502:
2501:
2496:
2476:
2475:
2463:
2440:
2420:
2405:
2401:
2399:
2398:
2393:
2379:
2378:
2362:
2354:
2343:distributive law
2340:
2333:
2322:
2311:
2296:
2286:
2275:
2273:
2272:
2267:
2262:
2261:
2256:
2243:
2221:
2217:
2213:
2209:
2186:
2180:
2169:
2158:
2154:
2140:
2136:
2134:
2133:
2128:
2108:
2107:
2091:
2085:
2079:
2072:
2061:
2055:
2049:
2038:
2032:
2024:
2018:
2010:
2004:
1987:
1979:
1969:
1961:
1955:
1947:
1936:
1930:
1924:
1916:
1914:
1913:
1908:
1897:
1874:
1854:
1836:
1830:
1822:
1813:
1805:
1798:
1792:
1779:Non-prime fields
1769:
1764:is a divisor of
1763:
1757:
1747:
1737:
1732:is a divisor of
1731:
1725:
1717:
1710:It follows that
1706:
1704:
1703:
1698:
1677:
1653:
1652:
1632:
1620:
1618:
1617:
1612:
1601:
1600:
1579:
1571:
1552:
1546:
1540:
1530:
1524:
1514:
1508:
1499:
1493:
1487:
1481:
1475:
1469:
1455:
1453:
1441:, implying that
1440:
1438:
1430:
1420:
1414:
1408:
1402:
1394:
1392:
1391:
1386:
1378:
1377:
1351:
1341:
1296:
1290:
1284:
1278:
1272:
1255:
1244:
1236:
1228:
1226:
1225:
1220:
1202:
1192:
1164:
1163:
1147:
1137:
1130:is in the field
1129:
1123:
1110:
1104:
1087:binomial theorem
1084:
1077:freshman's dream
1074:
1072:
1071:
1066:
1064:
1063:
1051:
1050:
1038:
1037:
1003:
997:
991:
981:
973:
967:
961:
955:
949:
941:
935:
916:
910:
903:
897:
891:
885:
875:
866:
860:
854:
839:
829:
821:
812:
810:
809:
804:
802:
794:
789:
776:
771:integers modulo
768:
758:
739:. This group is
730:
724:
711:
702:
694:
683:
681:
680:
675:
673:
672:
667:
640:
634:
621:
616:of the field is
611:
605:
599:
593:
587:
578:
572:
514:computer science
504:
498:
492:
471:
464:
419:
412:
405:
194:Commutative ring
123:Rack and quandle
88:
87:
81:
74:
70:
67:
61:
56:this article by
47:inline citations
34:
33:
26:
21:Galois extension
9887:
9886:
9882:
9881:
9880:
9878:
9877:
9876:
9862:
9861:
9852:
9847:
9824:
9794:
9776:
9758:
9748:Basic algebra I
9718:12(1): 22–38,
9708:
9703:
9698:
9697:
9682:
9674:. p. 123.
9672:Springer-Verlag
9664:
9660:
9613:
9609:
9595:
9563:
9559:
9553:
9531:
9527:
9522:
9518:
9510:
9506:
9498:
9487:
9481:
9480:
9476:
9463:
9459:
9445:
9438:
9433:
9391:
9357:
9324:
9320:
9319:
9314:
9313:
9311:
9308:
9307:
9290:
9286:
9284:
9281:
9280:
9260:
9255:
9242:
9238:
9236:
9233:
9232:
9204:
9201:
9200:
9181:
9177:
9176:
9171:
9170:
9168:
9165:
9164:
9147:
9143:
9141:
9138:
9137:
9113:
9109:
9108:
9103:
9102:
9100:
9097:
9096:
9079:
9071:
9065:
9062:
9061:
9036:
9032:
9031:
9026:
9025:
9023:
9020:
9019:
8999:
8995:
8993:
8990:
8989:
8983:
8958:
8948:
8946:
8945:
8943:
8940:
8939:
8934:of fields; Its
8909:
8905:
8904:
8899:
8898:
8886:
8873:
8863:
8861:
8860:
8858:
8855:
8854:
8832:
8829:
8828:
8806:
8802:
8801:
8796:
8795:
8784:
8780:
8779:
8773:
8772:
8771:
8769:
8766:
8765:
8737:
8733:
8732:
8728:
8726:
8723:
8722:
8703:
8699:
8698:
8693:
8692:
8690:
8687:
8686:
8679:
8654:
8649:
8648:
8646:
8643:
8642:
8625:
8615:
8613:
8612:
8610:
8607:
8606:
8602:
8593:
8587:
8576:
8570:
8529:
8502:
8499:
8498:
8492:
8491:A finite field
8489:
8453:
8448:
8389:Hasse principle
8355:one or several
8337:
8333:
8325:
8321:
8283:elliptic curves
8271:
8258:
8252:
8246:
8240:
8234:
8205:
8201:
8197:
8190:
8171:
8158:
8154:
8153:
8149:
8139:
8116:
8113:
8112:
8089:
8081:
8075:
8058:
8055:Möbius function
8048:
8024:
8020:
8016:
7992:
7978:
7955:
7952:
7951:
7943:
7937:
7922:
7919:
7900:
7887:
7877:
7871:
7861:
7853:
7847:
7839:
7833:
7827:
7821:
7813:
7807:
7799:
7796:splitting field
7789:
7779:
7771:
7765:
7759:
7751:
7741:
7731:
7722:
7716:
7693:
7689:
7687:
7684:
7683:
7682:The polynomial
7680:
7657:polynomial ring
7647:
7641:
7631:
7621:
7618:
7612:
7588:
7576:
7544:
7540:
7525:
7521:
7506:
7502:
7491:
7489:
7486:
7485:
7477:
7472:
7464:
7456:
7448:
7439:
7414:
7410:
7409:
7405:
7403:
7400:
7399:
7393:
7382:
7376:
7354:
7350:
7349:
7345:
7330:
7326:
7324:
7321:
7320:
7314:
7308:
7298:
7279:
7271:
7255:
7238:
7234:
7220:
7217:
7216:
7197:
7196:, the identity
7189:
7180:
7170:
7164:
7161:
7144:
7140:
7101:
7097:
7088:
7084:
7075:
7071:
7047:
7043:
7034:
7030:
7021:
7017:
6999:
6995:
6986:
6982:
6973:
6969:
6960:
6956:
6938:
6934:
6925:
6921:
6897:
6893:
6869:
6865:
6856:
6852:
6843:
6839:
6834:
6831:
6830:
6826:
6822:
6786:
6782:
6773:
6769:
6760:
6756:
6747:
6743:
6719:
6715:
6706:
6702:
6693:
6689:
6684:
6681:
6680:
6676:
6647:
6643:
6634:
6630:
6628:
6625:
6624:
6620:
6612:
6601:
6597:
6593:
6589:
6585:
6581:
6577:
6569:
6563:
6557:
6553:
6549:
6545:
6541:
6535:
6529:
6525:
6518:
6508:
6504:
6492:
6489:
6486:
6485:
6483:
6481:
6477:
6473:
6469:
6465:
6461:
6457:
6453:
6449:
6445:
6438:
6434:
6430:
6426:
6418:
6414:
6410:
6406:
6402:
6398:
6394:
6390:
6386:
6382:
6371:
6367:
6359:
6356:
6354:Example: GF(64)
6345:
6339:
6331:
6325:
6317:
6307:
6301:
6295:
6285:
6279:
6269:
6263:
6253:
6244:
6236:
6230:
6222:
6216:
6201:
6193:
6187:
6172:
6164:
6158:
6151:
6145:
6138:
6132:
6126:
6118:
6100:
6094:
6088:
6082:
6076:
6070:
6060:
6053:
6046:
6036:
6030:
6019:
6012:
6005:
5988:
5977:
5966:
5963:
5945:
5933:
5926:
5918:
5895:
5886:
5880:
5870:
5867:
5857:
5843:
5837:
5831:
5825:
5817:
5811:
5808:
5793:
5785:
5779:
5769:
5756:
5745:
5738:
5730:
5720:
5717:
5698:
5688:
5679:
5670:
5660:
5641:
5634:
5628:
5621:
5613:
5607:
5600:
5593:
5587:
5576:
5564:
5561:
5552:
5546:
5540:
5534:
5519:
5515:
5507:
5491:
5490:
5484:
5480:
5423:
5419:
5354:
5353:
5231:
5222:
5218:
5206:
5202:
5172:
5168:
5156:
5152:
5128:
5127:
5121:
5117:
5093:
5089:
5028:
5019:
5015:
5003:
4999:
4966:
4962:
4950:
4946:
4921:
4919:
4916:
4915:
4911:
4907:
4903:
4899:
4895:
4891:
4885:
4856:
4852:
4850:
4847:
4846:
4840:
4836:
4832:
4828:
4810:
4790:
4786:
4774:
4770:
4750:
4747:
4746:
4739:
4735:
4731:
4702:
4698:
4696:
4693:
4692:
4691:The polynomial
4689:
4671:
4670:
4664:
4660:
4527:
4518:
4514:
4484:
4480:
4456:
4455:
4449:
4445:
4384:
4375:
4371:
4338:
4334:
4310:
4309:
4304:
4284:
4279:
4272:
4268:
4222:
4213:
4209:
4181:
4179:
4176:
4175:
4171:
4167:
4163:
4159:
4155:
4151:
4120:
4116:
4114:
4111:
4110:
4104:
4100:
4096:
4082:
4062:
4058:
4038:
4035:
4034:
4027:
4023:
4019:
4015:
4011:
4007:
4003:
3999:
3970:
3966:
3964:
3961:
3960:
3959:The polynomial
3957:
3939:
3938:
3926:
3922:
3916:
3912:
3900:
3896:
3869:
3865:
3859:
3855:
3843:
3839:
3826:
3817:
3813:
3795:
3794:
3736:
3697:
3696:
3653:
3611:
3610:
3579:
3554:
3552:
3549:
3548:
3540:
3532:
3524:
3516:
3510:
3504:
3476:
3473:
3472:
3464:
3460:
3454:
3444:
3438:
3432:
3426:
3416:
3408:
3401:
3394:
3391:= 5, 7, 17, ...
3387:
3383:
3376:
3372:
3371:. For example,
3366:
3358:
3355:
3349:
3348:
3346:
3345:
3339:
3329:
3328:if and only if
3321:
3311:
3300:
3294:
3284:
3278:
3271:
3263:
3256:
3233:
3226:
3220:
3199:
3195:
3181:
3178:
3177:
3172:
3162:
3150:
3142:
3136:
3130:
3124:
3118:
3105:
3098:
3090:
3082:
3072:
3067:
3060:
3053:
3044:
3036:
3031:
3026:
3019:
3014:
3009:
3004:
2994:
2987:
2982:
2979:
2974:
2972:
2967:
2951:
2946:
2938:
2933:
2925:
2918:
2910:
2903:
2898:
2891:
2881:
2874:
2869:
2864:
2859:
2852:
2847:
2842:
2837:
2832:
2822:
2815:
2810:
2805:
2802:
2797:
2795:
2790:
2776:
2771:
2764:
2756:
2748:
2741:
2736:
2728:
2721:
2714:
2705:
2697:
2692:
2687:
2682:
2672:
2665:
2660:
2655:
2650:
2640:
2633:
2628:
2623:
2620:
2615:
2613:
2608:
2590:
2578:
2577:Multiplication
2566:
2554:
2550:
2546:
2539:
2533:
2530:
2520:
2514:
2508:
2471:
2467:
2459:
2433:
2413:
2411:
2408:
2407:
2403:
2402:Therefore, for
2374:
2370:
2368:
2365:
2364:
2360:
2352:
2335:
2324:
2313:
2298:
2288:
2277:
2257:
2252:
2251:
2249:
2246:
2245:
2241:
2238:
2219:
2215:
2211:
2192:
2182:
2171:
2160:
2156:
2142:
2138:
2103:
2099:
2097:
2094:
2093:
2087:
2081:
2077:
2063:
2057:
2051:
2040:
2034:
2026:
2020:
2012:
2006:
1998:
1981:
1975:
1963:
1957:
1949:
1941:
1932:
1926:
1918:
1893:
1867:
1847:
1845:
1842:
1841:
1832:
1824:
1818:
1807:
1800:
1794:
1784:
1781:
1776:
1765:
1759:
1758:if and only if
1749:
1739:
1733:
1727:
1726:if and only if
1719:
1711:
1667:
1648:
1644:
1642:
1639:
1638:
1624:
1596:
1592:
1590:
1587:
1586:
1575:
1567:
1548:
1542:
1532:
1526:
1516:
1510:
1504:
1495:
1489:
1483:
1477:
1471:
1465:
1451:
1442:
1436:
1432:
1426:
1416:
1410:
1404:
1396:
1373:
1369:
1361:
1358:
1357:
1354:splitting field
1347:
1333:
1330:
1292:
1286:
1280:
1274:
1268:
1261:field extension
1246:
1238:
1230:
1185:
1178:
1159:
1155:
1153:
1150:
1149:
1139:
1131:
1125:
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1094:
1080:
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1033:
1029:
1015:
1012:
999:
993:
987:
975:
969:
963:
957:
951:
943:
937:
918:
912:
905:
899:
893:
887:
877:
871:
862:
856:
850:
835:
823:
817:
798:
790:
785:
783:
780:
779:
772:
764:
754:
726:
716:
707:
696:
693:
685:
668:
663:
662:
660:
657:
656:
636:
626:
617:
607:
601:
595:
589:
583:
574:
568:
546:
530:finite geometry
500:
494:
488:
467:
460:
443:Évariste Galois
423:
394:
393:
392:
363:Non-associative
345:
334:
333:
323:
303:
292:
291:
280:Map of lattices
276:
272:Boolean algebra
267:Heyting algebra
241:
230:
229:
223:
204:Integral domain
168:
157:
156:
150:
104:
82:
71:
65:
62:
52:Please help to
51:
35:
31:
24:
17:
12:
11:
5:
9885:
9875:
9874:
9860:
9859:
9851:
9850:External links
9848:
9846:
9845:
9832:"Galois field"
9827:
9822:
9800:Lidl, Rudolf;
9797:
9792:
9779:
9774:
9761:
9756:
9740:
9726:
9704:
9702:
9699:
9696:
9695:
9680:
9658:
9607:
9593:
9557:
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9504:
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9435:
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9432:
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9417:
9412:
9407:
9402:
9397:
9390:
9387:
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9323:
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9268:
9263:
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9254:
9250:
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9208:
9184:
9180:
9174:
9150:
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9116:
9112:
9106:
9082:
9077:
9074:
9070:
9049:
9042:
9039:
9035:
9029:
9005:
9002:
8998:
8982:
8979:
8966:
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8955:
8951:
8919:
8912:
8908:
8902:
8895:
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8866:
8842:
8839:
8836:
8812:
8809:
8805:
8799:
8794:
8787:
8783:
8776:
8753:
8750:
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8706:
8702:
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8657:
8652:
8628:
8622:
8618:
8558:
8555:
8552:
8549:
8546:
8543:
8538:
8535:
8532:
8528:
8524:
8521:
8518:
8515:
8512:
8509:
8506:
8488:
8485:
8465:division rings
8452:
8449:
8447:
8444:
8377:Hensel lifting
8365:linear algebra
8287:Diffie–Hellman
8270:
8267:
8222:
8218:
8212:
8208:
8204:
8200:
8189:
8183:
8180:
8177:
8174:
8170:
8166:
8161:
8157:
8152:
8146:
8143:
8138:
8135:
8132:
8129:
8126:
8123:
8120:
8067:above and the
8036:
8031:
8027:
8023:
8019:
8015:
8012:
8009:
8006:
8001:
7998:
7995:
7991:
7985:
7982:
7977:
7974:
7971:
7968:
7965:
7962:
7959:
7918:
7915:
7764:. In fact, if
7704:
7701:
7696:
7692:
7679:
7676:
7614:Main article:
7611:
7608:
7599:Galois group.
7595:, which has a
7558:
7553:
7550:
7547:
7543:
7539:
7536:
7533:
7528:
7524:
7520:
7517:
7514:
7509:
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7501:
7497:
7494:
7427:
7424:
7417:
7413:
7408:
7364:
7357:
7353:
7348:
7344:
7341:
7338:
7333:
7329:
7241:
7237:
7233:
7230:
7227:
7224:
7179:is a power of
7160:
7157:
7137:
7136:
7124:
7121:
7118:
7115:
7112:
7109:
7104:
7100:
7096:
7091:
7087:
7083:
7078:
7074:
7070:
7067:
7064:
7061:
7058:
7055:
7050:
7046:
7042:
7037:
7033:
7029:
7024:
7020:
7016:
7013:
7010:
7007:
7002:
6998:
6994:
6989:
6985:
6981:
6976:
6972:
6968:
6963:
6959:
6955:
6952:
6949:
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6803:
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6789:
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6755:
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6709:
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6696:
6692:
6688:
6673:
6661:
6658:
6655:
6650:
6646:
6642:
6637:
6633:
6528:are primitive
6355:
6352:
6004:
6003:Roots of unity
6001:
5996:linear algebra
5944:
5875:is called the
5856:
5807:
5804:
5656:
5560:
5557:
5487:
5483:
5479:
5476:
5473:
5470:
5467:
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4638:
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4608:
4605:
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4599:
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4593:
4590:
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4569:
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4560:
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4483:
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4432:
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4399:
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4390:
4387:
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4364:
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4341:
4337:
4333:
4330:
4327:
4324:
4321:
4318:
4315:
4312:
4311:
4303:
4300:
4297:
4294:
4290:
4287:
4275:
4271:
4267:
4264:
4261:
4258:
4255:
4252:
4249:
4246:
4243:
4240:
4237:
4234:
4231:
4228:
4225:
4223:
4221:
4216:
4212:
4208:
4205:
4202:
4199:
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4193:
4190:
4187:
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4137:
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4128:
4123:
4119:
4070:
4065:
4061:
4057:
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3932:
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3919:
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3911:
3908:
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3889:
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3862:
3858:
3854:
3851:
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3838:
3835:
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3823:
3820:
3816:
3812:
3809:
3806:
3803:
3800:
3797:
3796:
3793:
3790:
3787:
3784:
3781:
3778:
3775:
3772:
3769:
3766:
3763:
3760:
3757:
3754:
3751:
3748:
3745:
3742:
3739:
3737:
3735:
3732:
3729:
3726:
3723:
3720:
3717:
3714:
3711:
3708:
3705:
3702:
3699:
3698:
3695:
3692:
3689:
3686:
3683:
3680:
3677:
3674:
3671:
3668:
3665:
3662:
3659:
3656:
3654:
3652:
3649:
3646:
3643:
3640:
3637:
3634:
3631:
3628:
3625:
3622:
3619:
3616:
3613:
3612:
3609:
3606:
3603:
3600:
3597:
3594:
3591:
3588:
3585:
3582:
3580:
3578:
3575:
3572:
3569:
3566:
3563:
3560:
3557:
3556:
3492:
3489:
3486:
3483:
3480:
3255:
3245:
3219:, which sends
3202:
3198:
3194:
3191:
3188:
3185:
3170:
3113:
3112:
3109:
3108:
3103:
3096:
3088:
3079:
3078:
3070:
3065:
3058:
3050:
3049:
3042:
3034:
3029:
3023:
3022:
3017:
3012:
3007:
3001:
3000:
2992:
2985:
2980:
2973:
2966:
2960:
2957:
2956:
2949:
2944:
2936:
2931:
2922:
2921:
2916:
2908:
2901:
2896:
2888:
2887:
2879:
2872:
2867:
2862:
2856:
2855:
2850:
2845:
2840:
2835:
2829:
2828:
2820:
2813:
2808:
2803:
2796:
2789:
2783:
2780:
2779:
2774:
2769:
2762:
2754:
2745:
2744:
2739:
2734:
2726:
2719:
2711:
2710:
2703:
2695:
2690:
2685:
2679:
2678:
2670:
2663:
2658:
2653:
2647:
2646:
2638:
2631:
2626:
2621:
2614:
2607:
2600:
2599:
2587:
2575:
2519:
2494:
2491:
2488:
2485:
2482:
2479:
2474:
2470:
2466:
2462:
2458:
2455:
2452:
2449:
2446:
2443:
2439:
2436:
2432:
2429:
2426:
2423:
2419:
2416:
2391:
2388:
2385:
2382:
2377:
2373:
2265:
2260:
2255:
2237:
2234:
2126:
2123:
2120:
2117:
2114:
2111:
2106:
2102:
1906:
1903:
1900:
1896:
1892:
1889:
1886:
1883:
1880:
1877:
1873:
1870:
1866:
1863:
1860:
1857:
1853:
1850:
1780:
1777:
1775:
1772:
1696:
1693:
1690:
1687:
1684:
1681:
1676:
1673:
1670:
1666:
1662:
1659:
1656:
1651:
1647:
1610:
1607:
1604:
1599:
1595:
1462:above identity
1384:
1381:
1376:
1372:
1368:
1365:
1329:
1326:
1316:(or sometimes
1218:
1215:
1212:
1209:
1206:
1201:
1198:
1195:
1191:
1188:
1184:
1181:
1177:
1173:
1170:
1167:
1162:
1158:
1062:
1058:
1054:
1049:
1045:
1041:
1036:
1032:
1028:
1025:
1022:
1019:
950:by an element
936:of an element
801:
797:
793:
788:
689:
671:
666:
614:characteristic
545:
542:
425:
424:
422:
421:
414:
407:
399:
396:
395:
391:
390:
385:
380:
375:
370:
365:
360:
354:
353:
352:
346:
340:
339:
336:
335:
332:
331:
328:Linear algebra
322:
321:
316:
311:
305:
304:
298:
297:
294:
293:
290:
289:
286:Lattice theory
282:
275:
274:
269:
264:
259:
254:
249:
243:
242:
236:
235:
232:
231:
222:
221:
216:
211:
206:
201:
196:
191:
186:
181:
176:
170:
169:
163:
162:
159:
158:
149:
148:
143:
138:
132:
131:
130:
125:
120:
111:
105:
99:
98:
95:
94:
84:
83:
38:
36:
29:
15:
9:
6:
4:
3:
2:
9884:
9873:
9872:Finite fields
9870:
9869:
9867:
9857:
9856:Finite Fields
9854:
9853:
9843:
9839:
9838:
9833:
9828:
9825:
9823:0-521-39231-4
9819:
9815:
9810:
9809:
9808:Finite Fields
9803:
9798:
9795:
9789:
9786:, CRC Press,
9785:
9780:
9777:
9771:
9767:
9762:
9759:
9753:
9749:
9745:
9741:
9739:
9735:
9731:
9727:
9725:
9721:
9717:
9711:
9706:
9705:
9691:
9687:
9683:
9677:
9673:
9669:
9662:
9654:
9650:
9645:
9640:
9635:
9630:
9626:
9622:
9618:
9611:
9604:
9600:
9596:
9594:9780511734885
9590:
9586:
9582:
9577:
9572:
9568:
9561:
9554:
9552:9783110283600
9548:
9544:
9540:
9536:
9529:
9520:
9513:
9512:Jacobson 2009
9508:
9497:
9493:
9486:
9485:
9478:
9472:, p. 10.
9471:
9467:
9461:
9453:
9449:
9443:
9441:
9436:
9426:
9425:Hamming space
9423:
9421:
9418:
9416:
9413:
9411:
9408:
9406:
9403:
9401:
9398:
9396:
9393:
9392:
9386:
9384:
9383:
9378:
9374:
9370:
9366:
9362:
9352:
9350:
9345:
9332:
9325:
9321:
9291:
9287:
9266:
9261:
9256:
9252:
9248:
9243:
9239:
9218:
9215:
9212:
9209:
9206:
9182:
9178:
9148:
9144:
9134:
9121:
9114:
9110:
9080:
9075:
9072:
9068:
9047:
9040:
9037:
9033:
9003:
9000:
8996:
8988:
8978:
8964:
8959:
8937:
8933:
8917:
8910:
8906:
8893:
8890:
8887:
8883:
8879:
8874:
8840:
8837:
8834:
8810:
8807:
8803:
8792:
8785:
8781:
8751:
8748:
8745:
8738:
8734:
8729:
8704:
8700:
8683:
8676:
8660:
8655:
8626:
8599:
8596:
8590:
8583:
8579:
8573:
8556:
8550:
8547:
8544:
8536:
8533:
8530:
8526:
8522:
8519:
8516:
8510:
8504:
8495:
8484:
8482:
8478:
8474:
8473:alternativity
8470:
8469:associativity
8466:
8462:
8458:
8457:division ring
8443:
8441:
8437:
8433:
8429:
8425:
8424:combinatorics
8420:
8418:
8417:character sum
8414:
8410:
8406:
8401:
8399:
8395:
8391:
8390:
8384:
8382:
8381:LLL algorithm
8378:
8374:
8370:
8366:
8362:
8358:
8357:prime numbers
8354:
8350:
8349:number theory
8345:
8343:
8331:
8319:
8315:
8311:
8306:
8304:
8303:vector spaces
8300:
8296:
8295:coding theory
8292:
8288:
8284:
8280:
8276:
8266:
8262:
8255:
8249:
8243:
8237:
8220:
8216:
8210:
8206:
8202:
8198:
8187:
8181:
8178:
8175:
8172:
8168:
8164:
8159:
8155:
8150:
8144:
8141:
8136:
8130:
8127:
8124:
8118:
8109:
8105:
8101:
8097:
8093:
8085:
8078:
8072:
8070:
8065:
8061:
8056:
8051:
8034:
8029:
8025:
8021:
8017:
8010:
8004:
7999:
7996:
7993:
7989:
7983:
7980:
7975:
7969:
7966:
7963:
7957:
7947:
7940:
7933:
7929:
7925:
7914:
7912:
7911:
7904:
7897:
7894:
7890:
7884:
7880:
7874:
7868:
7864:
7857:
7850:
7843:
7836:
7830:
7824:
7817:
7810:
7803:
7797:
7792:
7786:
7782:
7775:
7768:
7762:
7755:
7748:
7744:
7738:
7734:
7728:
7725:
7719:
7702:
7699:
7694:
7690:
7675:
7673:
7669:
7664:
7662:
7658:
7653:
7650:
7644:
7639:
7634:
7629:
7624:
7617:
7607:
7605:
7600:
7598:
7592:
7586:
7580:
7574:
7573:Galois theory
7569:
7556:
7551:
7548:
7545:
7541:
7537:
7534:
7531:
7526:
7522:
7518:
7515:
7512:
7507:
7503:
7499:
7481:
7475:
7468:
7460:
7452:
7445:
7442:
7425:
7422:
7415:
7411:
7406:
7396:
7390:
7386:
7379:
7362:
7355:
7351:
7346:
7339:
7336:
7331:
7327:
7317:
7311:
7306:
7301:
7295:
7293:
7289:
7283:
7275:
7269:
7265:
7259:
7239:
7235:
7228:
7225:
7222:
7213:
7209:
7205:
7201:
7193:
7186:
7183:
7177:
7173:
7167:
7156:
7152:
7148:
7122:
7116:
7113:
7110:
7107:
7102:
7098:
7094:
7089:
7085:
7081:
7076:
7072:
7062:
7059:
7056:
7053:
7048:
7044:
7040:
7035:
7031:
7027:
7022:
7018:
7008:
7005:
7000:
6996:
6992:
6987:
6983:
6979:
6974:
6970:
6966:
6961:
6957:
6947:
6944:
6939:
6935:
6931:
6926:
6922:
6912:
6909:
6906:
6903:
6898:
6894:
6884:
6881:
6878:
6875:
6870:
6866:
6862:
6857:
6853:
6849:
6844:
6840:
6820:
6817:
6801:
6795:
6792:
6787:
6783:
6779:
6774:
6770:
6766:
6761:
6757:
6753:
6748:
6744:
6734:
6731:
6728:
6725:
6720:
6716:
6712:
6707:
6703:
6699:
6694:
6690:
6674:
6659:
6656:
6653:
6648:
6644:
6640:
6635:
6631:
6618:
6617:
6616:
6610:
6605:
6575:
6566:
6560:
6538:
6532:
6522:
6515:
6511:
6502:
6444:The union of
6442:
6424:
6407:GF(2) = GF(8)
6403:GF(2) = GF(4)
6379:
6377:
6365:
6351:
6348:
6342:
6335:
6328:
6321:
6316:factors over
6315:
6310:
6304:
6298:
6294:
6288:
6282:
6277:
6272:
6266:
6261:
6256:
6250:
6247:
6240:
6233:
6226:
6219:
6213:
6209:
6205:
6197:
6190:
6185:
6179:
6175:
6168:
6161:
6154:
6148:
6141:
6135:
6129:
6122:
6115:
6112:
6108:
6104:
6097:
6091:
6085:
6079:
6073:
6067:
6063:
6056:
6049:
6044:
6039:
6033:
6027:
6023:
6015:
6010:
6009:root of unity
6000:
5997:
5992:
5984:
5980:
5975:
5969:
5960:
5956:
5952:
5948:
5943:
5940:
5936:
5929:
5922:
5915:
5914:for details.
5913:
5912:
5907:
5903:
5898:
5892:
5889:
5883:
5878:
5873:
5869:This integer
5864:
5860:
5855:
5851:
5847:
5840:
5834:
5828:
5821:
5814:
5803:
5801:
5796:
5789:
5782:
5776:
5772:
5766:
5764:
5759:
5752:
5748:
5741:
5734:
5728:
5723:
5713:
5709:
5705:
5701:
5697:
5692:
5687:
5682:
5678:
5677:such that the
5673:
5669:
5664:
5659:
5655:
5653:
5649:
5644:
5637:
5631:
5624:
5617:
5610:
5603:
5596:
5590:
5585:
5579:
5574:
5573:abelian group
5568:
5556:
5549:
5543:
5537:
5532:
5526:
5522:
5513:
5504:
5485:
5481:
5474:
5471:
5468:
5465:
5462:
5459:
5456:
5453:
5450:
5447:
5444:
5441:
5438:
5435:
5429:
5424:
5420:
5413:
5410:
5407:
5404:
5401:
5398:
5395:
5392:
5389:
5386:
5383:
5380:
5377:
5374:
5371:
5368:
5365:
5358:
5350:
5346:
5340:
5337:
5334:
5331:
5328:
5325:
5322:
5319:
5316:
5313:
5310:
5307:
5304:
5301:
5298:
5295:
5292:
5289:
5286:
5283:
5277:
5271:
5268:
5265:
5262:
5259:
5256:
5253:
5250:
5247:
5244:
5241:
5235:
5233:
5223:
5219:
5215:
5212:
5207:
5203:
5199:
5196:
5193:
5190:
5187:
5184:
5173:
5169:
5165:
5162:
5157:
5153:
5149:
5146:
5143:
5140:
5137:
5134:
5122:
5118:
5111:
5108:
5105:
5099:
5094:
5090:
5083:
5080:
5077:
5071:
5068:
5062:
5059:
5056:
5050:
5044:
5041:
5038:
5032:
5030:
5020:
5016:
5012:
5009:
5004:
5000:
4996:
4993:
4990:
4987:
4984:
4981:
4975:
4967:
4963:
4959:
4956:
4951:
4947:
4943:
4940:
4937:
4934:
4931:
4928:
4888:
4871:
4868:
4865:
4862:
4857:
4853:
4843:
4835:(elements of
4825:
4821:
4817:
4813:
4796:
4791:
4787:
4783:
4780:
4775:
4771:
4767:
4764:
4761:
4758:
4755:
4752:
4745:
4717:
4714:
4711:
4708:
4703:
4699:
4684:
4665:
4661:
4654:
4651:
4648:
4645:
4642:
4639:
4636:
4633:
4630:
4627:
4624:
4618:
4615:
4609:
4606:
4603:
4600:
4597:
4594:
4591:
4588:
4585:
4582:
4579:
4576:
4573:
4570:
4564:
4558:
4555:
4552:
4549:
4546:
4543:
4540:
4537:
4531:
4529:
4519:
4515:
4511:
4508:
4505:
4502:
4499:
4496:
4485:
4481:
4477:
4474:
4471:
4468:
4465:
4462:
4450:
4446:
4439:
4436:
4433:
4427:
4424:
4418:
4415:
4412:
4406:
4400:
4397:
4394:
4388:
4386:
4376:
4372:
4368:
4365:
4362:
4359:
4356:
4353:
4347:
4339:
4335:
4331:
4328:
4325:
4322:
4319:
4316:
4301:
4295:
4273:
4269:
4262:
4259:
4253:
4250:
4244:
4241:
4235:
4232:
4229:
4226:
4224:
4214:
4210:
4206:
4203:
4200:
4197:
4194:
4191:
4185:
4148:
4135:
4132:
4129:
4126:
4121:
4117:
4107:
4093:
4089:
4085:
4068:
4063:
4059:
4055:
4052:
4049:
4046:
4043:
4040:
4033:
3985:
3982:
3979:
3976:
3971:
3967:
3952:
3935:
3930:
3927:
3917:
3913:
3909:
3906:
3901:
3897:
3887:
3884:
3878:
3873:
3870:
3860:
3856:
3852:
3849:
3844:
3840:
3833:
3830:
3828:
3821:
3818:
3810:
3807:
3804:
3801:
3791:
3785:
3782:
3779:
3776:
3773:
3767:
3761:
3758:
3755:
3752:
3749:
3746:
3740:
3738:
3730:
3727:
3724:
3721:
3712:
3709:
3706:
3703:
3693:
3687:
3684:
3681:
3675:
3669:
3666:
3663:
3657:
3655:
3647:
3644:
3641:
3638:
3632:
3626:
3623:
3620:
3617:
3607:
3601:
3598:
3592:
3589:
3586:
3583:
3581:
3573:
3570:
3567:
3564:
3558:
3544:
3536:
3528:
3520:
3513:
3507:
3490:
3487:
3484:
3481:
3478:
3468:
3457:
3451:
3447:
3441:
3435:
3429:
3423:
3419:
3412:
3404:
3397:
3390:
3379:
3369:
3352:
3342:
3337:
3332:
3325:
3318:
3314:
3308:
3304:
3297:
3291:
3287:
3281:
3274:
3267:
3261:
3254:
3250:
3244:
3240:
3236:
3230:
3223:
3218:
3200:
3196:
3189:
3186:
3183:
3174:
3168:
3160:
3159:division by 0
3156:
3146:
3139:
3133:
3127:
3121:
3104:
3101:
3097:
3094:
3089:
3086:
3081:
3080:
3076:
3071:
3066:
3063:
3059:
3056:
3052:
3051:
3047:
3043:
3040:
3035:
3030:
3025:
3024:
3018:
3013:
3008:
3003:
3002:
2998:
2993:
2990:
2986:
2981:
2977:
2970:
2965:
2964:
2954:
2950:
2945:
2942:
2937:
2932:
2929:
2924:
2923:
2917:
2914:
2909:
2906:
2902:
2897:
2894:
2890:
2889:
2885:
2880:
2877:
2873:
2868:
2863:
2858:
2857:
2851:
2846:
2841:
2836:
2831:
2830:
2826:
2821:
2818:
2814:
2809:
2804:
2800:
2793:
2788:
2787:
2775:
2770:
2767:
2763:
2760:
2755:
2752:
2747:
2746:
2740:
2735:
2732:
2727:
2724:
2720:
2717:
2713:
2712:
2708:
2704:
2701:
2696:
2691:
2686:
2681:
2680:
2676:
2671:
2668:
2664:
2659:
2654:
2649:
2648:
2644:
2639:
2636:
2632:
2627:
2622:
2618:
2611:
2606:
2605:
2601:
2597:
2593:
2585:
2581:
2573:
2569:
2563:
2558:
2543:
2536:
2527:
2523:
2518:
2511:
2505:
2492:
2486:
2483:
2480:
2477:
2472:
2468:
2460:
2453:
2444:
2430:
2424:
2389:
2386:
2383:
2380:
2375:
2371:
2358:
2349:
2346:
2344:
2338:
2331:
2327:
2320:
2316:
2310:
2306:
2302:
2295:
2291:
2285:
2281:
2263:
2258:
2233:
2230:
2228:
2223:
2207:
2203:
2199:
2195:
2190:
2185:
2178:
2174:
2167:
2163:
2153:
2149:
2145:
2124:
2121:
2118:
2115:
2112:
2109:
2104:
2100:
2090:
2084:
2074:
2070:
2066:
2060:
2054:
2047:
2043:
2037:
2030:
2023:
2016:
2009:
2002:
1995:
1993:
1992:
1985:
1978:
1973:
1967:
1960:
1953:
1945:
1938:
1935:
1929:
1922:
1901:
1894:
1887:
1878:
1864:
1858:
1840:
1839:quotient ring
1835:
1828:
1821:
1817:
1811:
1803:
1797:
1791:
1787:
1771:
1768:
1762:
1756:
1752:
1746:
1742:
1736:
1730:
1723:
1715:
1707:
1694:
1688:
1685:
1682:
1674:
1671:
1668:
1664:
1660:
1657:
1654:
1649:
1645:
1636:
1635:
1631:
1627:
1623:
1608:
1605:
1602:
1597:
1593:
1584:
1583:
1578:
1574:
1570:
1566:
1561:
1559:
1554:
1551:
1545:
1539:
1535:
1529:
1523:
1519:
1513:
1507:
1501:
1498:
1492:
1486:
1480:
1474:
1470:are roots of
1468:
1463:
1459:
1450:
1446:
1435:
1429:
1424:
1419:
1413:
1407:
1400:
1382:
1379:
1374:
1370:
1366:
1363:
1355:
1350:
1345:
1340:
1336:
1325:
1323:
1319:
1315:
1314:division ring
1310:
1308:
1304:
1300:
1295:
1289:
1283:
1277:
1271:
1266:
1262:
1257:
1253:
1249:
1242:
1234:
1213:
1210:
1207:
1196:
1182:
1179:
1175:
1171:
1168:
1165:
1160:
1156:
1146:
1142:
1135:
1128:
1122:
1117:
1112:
1109:
1102:
1098:
1092:
1088:
1083:
1078:
1060:
1056:
1052:
1047:
1043:
1039:
1034:
1026:
1023:
1020:
1010:
1005:
1002:
996:
990:
985:
979:
972:
966:
960:
954:
947:
940:
934:
930:
926:
922:
915:
908:
902:
896:
890:
884:
880:
874:
870:
865:
859:
853:
847:
845:
844:
838:
833:
827:
820:
814:
795:
791:
777:
775:
767:
762:
757:
753:
748:
746:
742:
738:
734:
729:
723:
719:
715:
710:
704:
700:
692:
688:
669:
654:
650:
649:
644:
639:
633:
629:
623:
620:
615:
610:
604:
598:
592:
586:
582:
577:
571:
566:
562:
557:
555:
551:
541:
539:
538:coding theory
535:
531:
527:
526:Galois theory
523:
519:
518:number theory
515:
510:
508:
503:
497:
491:
486:
482:
477:
475:
470:
465:
463:
459:integers mod
456:
452:
448:
444:
440:
436:
432:
420:
415:
413:
408:
406:
401:
400:
398:
397:
389:
386:
384:
381:
379:
376:
374:
371:
369:
366:
364:
361:
359:
356:
355:
351:
348:
347:
343:
338:
337:
330:
329:
325:
324:
320:
317:
315:
312:
310:
307:
306:
301:
296:
295:
288:
287:
283:
281:
278:
277:
273:
270:
268:
265:
263:
260:
258:
255:
253:
250:
248:
245:
244:
239:
234:
233:
228:
227:
220:
217:
215:
214:Division ring
212:
210:
207:
205:
202:
200:
197:
195:
192:
190:
187:
185:
182:
180:
177:
175:
172:
171:
166:
161:
160:
155:
154:
147:
144:
142:
139:
137:
136:Abelian group
134:
133:
129:
126:
124:
121:
119:
115:
112:
110:
107:
106:
102:
97:
96:
93:
90:
89:
80:
77:
69:
66:February 2015
59:
55:
49:
48:
42:
37:
28:
27:
22:
9835:
9807:
9783:
9765:
9747:
9729:
9709:
9667:
9661:
9624:
9620:
9610:
9576:math/0409420
9566:
9560:
9534:
9528:
9519:
9507:
9483:
9477:
9460:
9451:
9448:Moore, E. H.
9415:Finite group
9380:
9358:
9346:
9135:
8984:
8936:direct limit
8932:directed set
8684:
8600:
8594:
8588:
8581:
8577:
8571:
8493:
8490:
8454:
8428:Paley Graphs
8421:
8402:
8387:
8385:
8346:
8307:
8275:cryptography
8272:
8269:Applications
8260:
8253:
8247:
8241:
8235:
8110:
8103:
8099:
8095:
8091:
8083:
8076:
8073:
8063:
8059:
8049:
7950:is given by
7945:
7938:
7931:
7927:
7923:
7920:
7908:
7902:
7898:
7892:
7888:
7882:
7878:
7872:
7866:
7862:
7855:
7848:
7841:
7834:
7828:
7822:
7815:
7808:
7801:
7790:
7784:
7780:
7773:
7766:
7760:
7753:
7746:
7742:
7736:
7732:
7729:
7723:
7717:
7681:
7665:
7654:
7648:
7642:
7632:
7622:
7619:
7601:
7590:
7578:
7571:In terms of
7570:
7479:
7473:
7471:has exactly
7466:
7458:
7450:
7446:
7440:
7394:
7388:
7384:
7377:
7315:
7313:with itself
7309:
7299:
7297:Denoting by
7296:
7281:
7273:
7257:
7211:
7207:
7203:
7199:
7191:
7187:
7181:
7175:
7171:
7165:
7162:
7150:
7146:
7138:
6606:
6564:
6558:
6536:
6530:
6523:
6513:
6509:
6499:irreducible
6460:elements of
6443:
6380:
6376:Galois group
6357:
6346:
6340:
6333:
6326:
6319:
6308:
6302:
6296:
6293:discriminant
6286:
6280:
6270:
6264:
6254:
6251:
6245:
6238:
6231:
6224:
6217:
6214:
6207:
6203:
6195:
6188:
6177:
6173:
6166:
6159:
6152:
6146:
6139:
6133:
6127:
6120:
6116:
6110:
6106:
6102:
6095:
6089:
6083:
6077:
6071:
6065:
6061:
6054:
6047:
6042:
6037:
6031:
6028:
6021:
6013:
6006:
5993:
5982:
5978:
5967:
5964:
5958:
5954:
5950:
5946:
5938:
5934:
5927:
5920:
5916:
5909:
5896:
5893:
5887:
5885:to the base
5881:
5871:
5868:
5862:
5858:
5849:
5845:
5838:
5832:
5826:
5819:
5812:
5809:
5798:is prime is
5794:
5787:
5780:
5774:
5770:
5767:
5757:
5750:
5746:
5739:
5732:
5725:is called a
5721:
5718:
5711:
5707:
5703:
5699:
5695:
5690:
5685:
5680:
5676:
5671:
5667:
5662:
5657:
5642:
5635:
5629:
5627:has at most
5622:
5615:
5608:
5601:
5594:
5588:
5577:
5566:
5562:
5547:
5541:
5535:
5524:
5520:
5505:
4886:
4841:
4823:
4819:
4815:
4811:
4690:
4149:
4105:
4091:
4087:
4083:
3958:
3542:
3534:
3526:
3518:
3511:
3505:
3466:
3455:
3449:
3445:
3439:
3433:
3427:
3424:
3417:
3410:
3402:
3395:
3388:
3377:
3367:
3350:
3340:
3330:
3323:
3316:
3312:
3309:
3302:
3295:
3289:
3285:
3279:
3272:
3265:
3257:
3252:
3248:
3238:
3234:
3228:
3221:
3175:
3144:
3137:
3131:
3125:
3119:
3116:
3099:
3092:
3084:
3074:
3061:
3054:
3045:
3038:
2996:
2988:
2975:
2968:
2952:
2940:
2927:
2912:
2904:
2892:
2883:
2875:
2824:
2816:
2798:
2791:
2765:
2758:
2750:
2730:
2722:
2715:
2706:
2699:
2674:
2666:
2642:
2634:
2616:
2609:
2595:
2591:
2583:
2579:
2571:
2567:
2541:
2534:
2531:
2525:
2521:
2509:
2506:
2350:
2347:
2336:
2334:, for every
2329:
2325:
2318:
2314:
2308:
2304:
2300:
2293:
2289:
2283:
2279:
2239:
2231:
2224:
2205:
2201:
2197:
2193:
2183:
2176:
2172:
2165:
2161:
2151:
2147:
2143:
2088:
2082:
2075:
2068:
2064:
2058:
2052:
2045:
2041:
2035:
2028:
2021:
2014:
2007:
2000:
1996:
1989:
1983:
1976:
1965:
1958:
1951:
1943:
1939:
1933:
1927:
1920:
1833:
1826:
1819:
1809:
1806:, the field
1801:
1795:
1789:
1785:
1782:
1766:
1760:
1754:
1750:
1744:
1740:
1734:
1728:
1721:
1713:
1709:
1637:
1633:
1629:
1625:
1621:
1585:
1581:
1576:
1572:
1568:
1564:
1563:
1555:
1549:
1543:
1537:
1533:
1527:
1521:
1517:
1511:
1505:
1502:
1496:
1490:
1484:
1478:
1472:
1466:
1448:
1444:
1433:
1427:
1417:
1411:
1405:
1398:
1348:
1338:
1334:
1331:
1317:
1311:
1293:
1287:
1281:
1275:
1269:
1258:
1251:
1247:
1240:
1232:
1144:
1140:
1133:
1126:
1120:
1113:
1107:
1100:
1096:
1081:
1006:
1000:
994:
988:
984:vector space
977:
970:
964:
958:
952:
945:
938:
932:
928:
924:
920:
913:
906:
900:
894:
888:
882:
878:
876:, denote by
872:
863:
857:
851:
848:
841:
836:
825:
818:
815:
773:
765:
755:
752:prime number
749:
727:
721:
717:
708:
705:
698:
690:
686:
646:
637:
631:
627:
624:
618:
608:
602:
596:
590:
584:
575:
569:
564:
560:
558:
554:field axioms
547:
534:cryptography
516:, including
511:
501:
495:
489:
480:
478:
474:prime number
468:
461:
439:Galois field
438:
435:finite field
434:
428:
388:Hopf algebra
326:
319:Vector space
284:
224:
153:Group theory
151:
116: /
72:
63:
44:
9627:: 233–274.
9466:E. H. Moore
9410:Finite ring
8419:estimates.
8413:exponential
8192: prime
7921:The number
7638:irreducible
7305:composition
6570:{9, 21, 63}
6413:itself. As
6330:, and that
6125:contains a
4827:are either
4744:expressions
4032:expressions
2222:as a root.
1558:E. H. Moore
1344:prime power
1259:Any finite
886:the sum of
761:prime field
581:prime power
485:prime power
431:mathematics
373:Lie algebra
358:Associative
262:Total order
252:Semilattice
226:Ring theory
58:introducing
9701:References
9690:1213.51001
9375:proved by
8446:Extensions
8312:, such as
8245:and every
7852:belong to
7820:of degree
6816:reciprocal
6604:elements.
6596:primitive
6588:primitive
6580:primitive
6503:of degree
6391:1, 2, 3, 6
6366:of degree
6358:The field
6117:The field
5981:= 0, ...,
5854:such that
5778:for every
5599:such that
5529:and their
5510:has eight
5506:The field
4884:(that is,
4281:(for
3400:, that is
3149:for every
2359:of degree
2328:⋅ 0 = 0 ⋅
2287:such that
2189:trinomials
1831:of degree
1799:prime and
1634:factors as
1318:skew field
1089:, as each
904:such that
892:copies of
714:polynomial
643:isomorphic
544:Properties
507:isomorphic
41:references
9842:EMS Press
9746:(2009) ,
9653:1071-5797
9377:Chevalley
8954:¯
8891:≥
8884:⋃
8869:¯
8793:⊂
8746:−
8621:¯
8551:α
8548:−
8534:∈
8531:α
8527:∏
8471:axiom to
8299:subspaces
8211:ℓ
8188:ℓ
8176:∣
8173:ℓ
8169:∑
8165:−
8137:≥
8005:μ
7997:∣
7990:∑
7826:dividing
7794:, as its
7700:−
7655:As every
7549:−
7542:φ
7535:…
7523:φ
7516:φ
7504:φ
7423:−
7343:↦
7328:φ
7232:↦
7223:φ
7145:GF(2) / (
6540:dividing
6464:generate
6452:has thus
5972:, called
5737:. Unless
5482:α
5421:α
5347:α
5220:α
5204:α
5194:α
5170:α
5154:α
5144:α
5119:α
5091:α
5069:α
5017:α
5001:α
4991:α
4964:α
4948:α
4938:α
4866:α
4854:α
4788:α
4772:α
4762:α
4662:α
4616:α
4516:α
4506:α
4482:α
4472:α
4447:α
4425:α
4373:α
4363:α
4336:α
4326:α
4270:α
4260:−
4251:α
4242:−
4230:−
4211:α
4201:α
4186:−
4130:α
4118:α
4060:α
4050:α
3983:−
3977:−
3936:α
3928:−
3907:−
3885:−
3871:−
3850:−
3819:−
3811:α
3792:α
3731:α
3713:α
3694:α
3648:α
3627:α
3608:α
3599:−
3587:−
3574:α
3559:−
3488:α
3398:≡ 3 mod 4
3193:↦
3184:φ
3153:in every
2589:Division
2565:Addition
2532:and that
2218:, having
1686:−
1672:∈
1665:∏
1655:−
1531:roots of
1380:−
1303:separable
1265:separable
1211:−
1183:∈
1176:∏
1166:−
763:of order
653:subfields
606:, adding
383:Bialgebra
189:Near-ring
146:Lie group
114:Semigroup
9866:Category
9804:(1997),
9603:28297089
9496:archived
9389:See also
9231:one has
8985:Given a
8586:for all
8580: (
8575:, since
8318:BCH code
7876:divides
7290:, after
6291:, whose
6221:, every
3176:The map
2039:, where
1748:divides
1009:identity
867:and any
824:0, ...,
725:has all
451:elements
219:Lie ring
184:Semiring
9514:, §4.13
9373:Dickson
8379:or the
8334:GF(929)
8053:is the
7870:; thus
7604:perfect
7444:roots.
7383:0 <
6496:
6484:
6423:coprime
6260:coprime
6109:, ...,
6087:, then
5706:, ...,
4839:), and
4018:nor in
3362:
3347:
3338:modulo
3293:, with
2339:∈ GF(4)
1352:be the
1307:perfect
1285:, then
974:into a
909:⋅ 1 = 0
869:integer
588:(where
445:) is a
350:Algebra
342:Algebra
247:Lattice
238:Lattice
54:improve
9820:
9790:
9772:
9754:
9688:
9678:
9651:
9601:
9591:
9549:
9279:where
8605:, let
8353:modulo
8330:PDF417
8185:
8047:where
7942:over
7907:; see
7597:cyclic
7266:and a
7141:GF(64)
6827:GF(64)
6526:GF(64)
6466:GF(64)
6462:GF(64)
6435:GF(64)
6411:GF(64)
6409:, and
6395:GF(64)
6389:being
6360:GF(64)
5976:, for
5908:, see
5894:While
5755:where
5742:= 2, 3
5652:cyclic
5646:. The
5571:is an
5516:GF(16)
5508:GF(16)
4904:GF(16)
4900:GF(16)
4809:where
4740:GF(16)
4687:GF(16)
4156:GF(27)
4081:where
4028:GF(27)
3431:, let
3382:, and
2524:= 1 +
2323:, and
2307:⋅ 1 =
2292:= 1 +
2282:, 1 +
2278:0, 1,
1804:> 1
1541:, and
1346:, and
1320:). By
759:, the
741:cyclic
712:, the
378:Graded
309:Module
300:Module
199:Domain
118:Monoid
43:, but
9712:≤ 169
9599:S2CID
9571:arXiv
9499:(PDF)
9488:(PDF)
9431:Notes
9379:(see
9369:Artin
9060:then
8675:up to
8584:) = 1
8434:. In
8326:GF(2)
8291:ECDHE
8257:over
8080:over
7886:. As
7740:then
7640:over
7583:is a
7387:<
7254:is a
6613:GF(2)
6611:over
6550:GF(8)
6546:GF(4)
6519:GF(2)
6517:over
6478:GF(2)
6474:GF(2)
6472:over
6450:GF(8)
6446:GF(4)
6439:GF(2)
6431:GF(8)
6427:GF(4)
6399:GF(2)
6372:GF(2)
6370:over
6144:; if
6075:is a
6069:. If
6064:<
6011:, as
5842:with
5582:. By
5545:with
4912:GF(2)
4908:GF(2)
4892:GF(2)
4837:GF(2)
4732:GF(2)
4172:GF(3)
4168:GF(2)
4164:GF(3)
4160:GF(2)
4152:GF(8)
4101:GF(3)
4097:GF(2)
4024:GF(8)
4020:GF(3)
4016:GF(2)
4004:GF(3)
4000:GF(2)
3503:with
3409:−1 ≡
3393:. If
3334:is a
3163:GF(4)
2555:GF(4)
2551:GF(2)
2547:GF(4)
2515:GF(4)
2404:GF(4)
2353:GF(2)
2351:Over
2242:GF(4)
2078:GF(4)
2071:) = 0
2048:) = 0
1793:with
1454:) = 1
1342:be a
1138:then
1118:, if
733:roots
645:(see
579:is a
561:order
550:field
481:order
472:is a
466:when
447:field
344:-like
302:-like
240:-like
209:Field
167:-like
141:Magma
109:Group
103:-like
101:Group
9818:ISBN
9788:ISBN
9770:ISBN
9752:ISBN
9676:ISBN
9649:ISSN
9589:ISBN
9547:ISBN
9371:and
8838:>
8827:for
8415:and
8403:The
8363:and
8094:− 1)
7303:the
7206:) =
7153:+ 1)
6821:The
6548:and
6482:9 =
6448:and
6429:and
6421:are
6417:and
6397:are
6210:− 1)
6202:gcd(
5961:+ 1)
5844:0 ≤
5753:− 1)
4154:and
4026:and
4010:and
4002:and
3509:and
3227:1 +
3157:the
3155:ring
3143:0 ⋅
3091:1 +
3083:1 +
3073:1 +
3037:1 +
2995:1 +
2939:1 +
2926:1 +
2911:1 +
2882:1 +
2823:1 +
2757:1 +
2749:1 +
2729:1 +
2698:1 +
2673:1 +
2641:1 +
2540:1 +
2538:and
2507:Let
2299:1 ⋅
2208:+ 1
1443:gcd(
1439:= −1
1415:has
1332:Let
1007:The
927:) ↦
849:Let
641:are
565:size
536:and
479:The
433:, a
174:Ring
165:Ring
9734:doi
9720:doi
9714:",
9686:Zbl
9639:hdl
9629:doi
9581:doi
9539:doi
9385:).
9163:of
9018:of
8938:is
8592:in
8344:.
8316:or
8301:of
8273:In
8259:GF(
8088:is
8082:GF(
7944:GF(
7901:GF(
7854:GF(
7840:GF(
7814:GF(
7800:GF(
7778:of
7772:GF(
7752:GF(
7636:is
7620:If
7589:GF(
7587:of
7577:GF(
7478:GF(
7465:GF(
7457:GF(
7449:GF(
7307:of
7280:GF(
7272:GF(
7270:of
7256:GF(
7190:GF(
7188:In
6568:in
6433:in
6332:GF(
6318:GF(
6312:th
6289:− 1
6274:th
6262:to
6258:is
6200:is
6194:GF(
6171:is
6165:GF(
6155:− 1
6142:− 1
6119:GF(
6101:1,
6057:= 1
6050:= 1
6041:th
6029:If
6020:GF(
6016:= 1
5991:).
5985:− 2
5970:+ 1
5930:– 1
5919:GF(
5879:of
5852:− 2
5830:in
5818:GF(
5810:If
5786:GF(
5784:in
5761:is
5731:GF(
5729:of
5714:= 1
5696:are
5689:GF(
5683:– 1
5661:GF(
5638:– 1
5625:= 1
5614:GF(
5612:in
5604:= 1
5597:– 1
5592:of
5580:– 1
5565:GF(
5527:+ 1
4894:is
4831:or
4170:or
4162:or
4099:or
3547:):
3541:GF(
3533:GF(
3525:GF(
3517:GF(
3515:in
3465:GF(
3420:+ 1
3413:− 1
3353:− 1
3322:GF(
3301:GF(
3299:in
3275:= 2
3264:GF(
3247:GF(
3241:+ 1
3147:= 0
3123:by
2332:= 0
2321:= 0
2244:or
2204:+
2200:+
2179:+ 1
2168:+ 1
2027:GF(
2013:GF(
1999:GF(
1982:GF(
1974:by
1964:GF(
1950:GF(
1942:GF(
1919:GF(
1825:GF(
1823:in
1808:GF(
1720:GF(
1712:GF(
1431:is
1425:of
1397:GF(
1301:is
1254:= 0
1239:GF(
1231:GF(
1132:GF(
1114:By
992:is
976:GF(
956:of
944:GF(
942:of
861:in
846:).
834:by
828:− 1
697:GF(
695:or
625:If
455:set
437:or
429:In
179:Rng
9868::
9840:,
9834:,
9816:,
9684:.
9647:.
9637:.
9625:32
9623:.
9619:.
9597:,
9587:,
9579:,
9545:,
9490:,
9439:^
9351:.
8841:1.
8682:.
8598:.
8483:.
8455:A
8383:.
8375:,
8265:.
8108:.
8102:,
8071:.
8062:−
7930:,
7913:.
7891:−
7881:−
7865:−
7783:−
7745:−
7735:=
7727:.
7652:.
7606:.
7294:.
7210:+
7202:+
7185:.
7174:=
7149:+
6823:36
6677:21
6602:54
6598:63
6594:36
6590:21
6586:12
6572:.
6554:54
6542:63
6521:.
6512:−
6487:54
6458:54
6454:10
6441:.
6405:,
6401:,
6378:.
6249:.
6239:np
6225:np
6212:.
6206:,
6114:.
6105:,
6026:.
5989:−∞
5953:=
5949:+
5937:+
5891:.
5861:=
5848:≤
5802:.
5773:=
5765:.
5710:,
5702:,
5675:,
5553:15
5523:+
4914:.
4822:,
4818:,
4814:,
4136:1.
4090:,
4086:,
3461:−1
3448:=
3422:.
3315:−
3307:.
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914:p
907:n
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895:x
889:n
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864:F
858:x
852:F
837:p
826:p
819:p
800:Z
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792:/
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766:p
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728:q
722:X
718:X
709:q
701:)
699:q
691:q
687:F
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638:q
632:p
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619:p
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597:k
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418:e
411:t
404:v
79:)
73:(
68:)
64:(
50:.
23:.
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