7170:
7130:
1276:, provided a proof for Fermat's Last Theorem. Almost every mathematician at the time had previously considered both Fermat's Last Theorem and the Modularity Theorem either impossible or virtually impossible to prove, even given the most cutting-edge developments. Wiles first announced his proof in June 1993 in a version that was soon recognized as having a serious gap at a key point. The proof was corrected by Wiles, partly in collaboration with
655:
7150:
7140:
7160:
6803:
5730:
gluing together local data. This spirit is adopted in algebraic number theory. Given a prime in the ring of algebraic integers in a number field, it is desirable to study the field locally at that prime. Therefore, one localizes the ring of algebraic integers to that prime and then completes the fraction field much in the spirit of geometry.
5002:
5729:
can be deduced easily from the analogous local statement. The philosophy behind the study of local fields is largely motivated by geometric methods. In algebraic geometry, it is common to study varieties locally at a point by localizing to a maximal ideal. Global information can then be recovered by
1655:
However, even with this weaker definition, many rings of integers in algebraic number fields do not admit unique factorization. There is an algebraic obstruction called the ideal class group. When the ideal class group is trivial, the ring is a UFD. When it is not, there is a distinction between a
924:
where he claimed he had a proof that was too large to fit in the margin. No successful proof was published until 1995 despite the efforts of countless mathematicians during the 358 intervening years. The unsolved problem stimulated the development of algebraic number theory in the 19th century and
2416:
Unique factorization fails if and only if there are prime ideals that fail to be principal. The object which measures the failure of prime ideals to be principal is called the ideal class group. Defining the ideal class group requires enlarging the set of ideals in a ring of algebraic integers so
6084:
states that the Artin symbol from ideals (or ideles) to elements of a Galois group is trivial on a certain subgroup. Several more recent generalizations express reciprocity laws using cohomology of groups or representations of adelic groups or algebraic K-groups, and their relationship with the
971:
was published, number theory consisted of a collection of isolated theorems and conjectures. Gauss brought the work of his predecessors together with his own original work into a systematic framework, filled in gaps, corrected unsound proofs, and extended the subject in numerous ways.
995:. Many of the annotations given by Gauss are in effect announcements of further research of his own, some of which remained unpublished. They must have appeared particularly cryptic to his contemporaries; we can now read them as containing the germs of the theories of
3067:
1095:"Although the book is assuredly based on Dirichlet's lectures, and although Dedekind himself referred to the book throughout his life as Dirichlet's, the book itself was entirely written by Dedekind, for the most part after Dirichlet's death." (Edwards 1983)
6000:
5433:
3918:
of the number field. The adele ring allows one to simultaneously track all the data available using absolute values. This produces significant advantages in situations where the behavior at one place can affect the behavior at other places, as in the
2387:
is prime, provides a complete description of the prime ideals in the
Gaussian integers. Generalizing this simple result to more general rings of integers is a basic problem in algebraic number theory. Class field theory accomplishes this goal when
4748:
2309:
2060:
is the principal ideal generated by a single element. This is the strongest sense in which the ring of integers of a general number field admits unique factorization. In the language of ring theory, it says that rings of integers are
2854:
2014:
3475:
3399:
1695:
admits a factorization into irreducible elements, but it may admit more than one. This is because, while all prime elements are irreducible, some irreducible elements may not be prime. For example, consider the ring
1148:, he used an existence proof that shows there must be solutions for the problem rather than providing a mechanism to produce the answers. He then had little more to publish on the subject; but the emergence of
3569:
2950:
1808:
1380:
is not a prime number because it is negative, but it is a prime element. If factorizations into prime elements are permitted, then, even in the integers, there are alternative factorizations such as
2601:
2126:
An ideal which is prime in the ring of integers in one number field may fail to be prime when extended to a larger number field. Consider, for example, the prime numbers. The corresponding ideals
5154:
2987:
4753:
4106:
5550:
2622:, form a subgroup of the group of all non-zero fractional ideals. The quotient of the group of non-zero fractional ideals by this subgroup is the ideal class group. Two fractional ideals
244:
1637:
1446:
3817:. Because absolute values are unable to distinguish between a complex embedding and its conjugate, a complex embedding and its conjugate determine the same place. Therefore, there are
536:
1085:'s study of Lejeune Dirichlet's work was what led him to his later study of algebraic number fields and ideals. In 1863, he published Lejeune Dirichlet's lectures on number theory as
5723:
5894:
5297:
4533:
3664:
2048:
4219:
3964:
5033:
4997:{\displaystyle {\begin{aligned}&a_{-k}(t-2)^{-k}+\cdots +a_{-1}(t-2)^{-1}+a_{0}+a_{1}(t-2)+a_{2}(t-2)^{2}+\cdots \\&=\sum _{n=-k}^{\infty }a_{n}(t-2)^{n}\end{aligned}}}
851:
489:
452:
3787:
3763:
3909:
955:
in 1798 when Gauss was 21 and first published in 1801 when he was 24. In this book Gauss brings together results in number theory obtained by mathematicians such as Fermat,
3877:
2654:. Therefore, the ideal class group makes two fractional ideals equivalent if one is as close to being principal as the other is. The ideal class group is generally denoted
4141:
198:
6119:. Via the analogy of function fields vs. number fields, it relies on techniques and ideas from algebraic geometry. Moreover, the study of higher-dimensional schemes over
5725:
a complete, discrete valued field with finite residue field. This process simplifies the arithmetic of the field and allows the local study of problems. For example, the
4498:
2091:
Historically, the idea of factoring ideals into prime ideals was preceded by Ernst Kummer's introduction of ideal numbers. These are numbers lying in an extension field
4407:
1508:
are associate, but only one of these is positive. Requiring that prime numbers be positive selects a unique element from among a set of associated prime elements. When
5076:
4710:
798:
4361:
4042:
2168:
1253:
found evidence supporting it, yet no proof; as a result the "astounding" conjecture was often known as the
Taniyama–Shimura-Weil conjecture. It became a part of the
5220:
therefore implies that it is a direct sum of a torsion part and a free part. Reinterpreting this in the context of a number field, the torsion part consists of the
4669:
4560:
4335:
5217:
4622:
4593:
4248:
4740:
3992:
4465:
4642:
4427:
4308:
4288:
4268:
4181:
4161:
4062:
1284:
and number theory, and has many ramifications in these branches of mathematics. It also uses standard constructions of modern algebraic geometry, such as the
2701:
is 2. This means that there are only two ideal classes, the class of principal fractional ideals, and the class of a non-principal fractional ideal such as
1648:, it is not true that factorizations are unique up to the order of the factors. For this reason, one adopts the definition of unique factorization used in
1190:
in a series of papers (1924; 1927; 1930). This law is a general theorem in number theory that forms a central part of global class field theory. The term "
6016:
There are several different ways to express reciprocity laws. The early reciprocity laws found in the 19th century were usually expressed in terms of a
2772:
746:, who studied them and developed methods for the solution of some kinds of Diophantine equations. A typical Diophantine problem is to find two integers
1912:. Unlike the situation with units, where uniqueness could be repaired by weakening the definition, overcoming this failure requires a new perspective.
1938:
6232:
3404:
1115:
that satisfy polynomial equations with integer coefficients. The concept underwent further development in the hands of
Hilbert and, especially, of
3334:
2119:. A generator of this principal ideal is called an ideal number. Kummer used these as a substitute for the failure of unique factorization in
1027:). The formula, which Jacobi called a result "touching the utmost of human acumen", opened the way for similar results regarding more general
2074:
is a UFD, every prime ideal is generated by a prime element. Otherwise, there are prime ideals which are not generated by prime elements. In
1280:, and the final, widely accepted version was released in September 1994, and formally published in 1995. The proof uses many techniques from
641:
1074:, for which he found the first results, is still an unsolved problem in number theory despite later contributions by other researchers.
2344:
7153:
2499:
is also a fractional ideal. This operation makes the set of non-zero fractional ideals into a group. The group identity is the ideal
6807:
5556:
of the number field. One of the simplifications made possible by working with the adele ring is that there is a single object, the
6853:
3521:
2872:
1652:(UFDs). In a UFD, the prime elements occurring in a factorization are only expected to be unique up to units and their ordering.
1372:. This property is closely related to primality in the integers, because any positive integer satisfying this property is either
856:
Diophantine equations have been studied for thousands of years. For example, the solutions to the quadratic
Diophantine equation
4007:
1730:
6769:
6730:
6634:
6607:
6580:
6513:
6390:
6275:
6213:
3168:, is so-called because it admits two real embeddings but no complex embeddings. These are the field homomorphisms which send
3062:{\displaystyle 1\to O^{\times }\to K^{\times }{\xrightarrow {\text{div}}}\operatorname {Div} K\to \operatorname {Cl} K\to 1.}
2522:
1055:
98:
5084:
685:, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic objects such as
6295:
2123:. These eventually led Richard Dedekind to introduce a forerunner of ideals and to prove unique factorization of ideals.
1317:, and this factorization is unique up to the ordering of the factors. This may no longer be true in the ring of integers
17:
4067:
5483:
6693:
6374:
6303:
2343:
are the same. A complete answer to the question of which ideals remain prime in the
Gaussian integers is provided by
1310:
634:
586:
1087:
7194:
6761:
6128:
5846:
5743:
One of the classical results in algebraic number theory is that the ideal class group of an algebraic number field
5172:. Other rings of integers may admit more units. The Gaussian integers have four units, the previous two as well as
3931:
There is a geometric analogy for places at infinity which holds on the function fields of curves. For example, let
208:
6365:
1547:
1541:, but there is no way to single out one as being more canonical than the other. This leads to equations such as
1386:
1249:
It was initially dismissed as unlikely or highly speculative, but was taken more seriously when number theorist
6884:
5995:{\displaystyle \left({\frac {p}{q}}\right)\left({\frac {q}{p}}\right)=(-1)^{{\frac {p-1}{2}}{\frac {q-1}{2}}}.}
3491:
1277:
1012:
988:
503:
5428:{\displaystyle {\begin{cases}L:K^{\times }\to \mathbf {R} ^{r_{1}+r_{2}}\\L(x)=(\log |x|_{v})_{v}\end{cases}}}
3674:
Real and complex embeddings can be put on the same footing as prime ideals by adopting a perspective based on
1111:, does not appear in Dedekind's work.) Dedekind defined an ideal as a subset of a set of numbers, composed of
738:
The beginnings of algebraic number theory can be traced to
Diophantine equations, named after the 3rd-century
6824:
6814:
6718:
5681:
3722:(up to equivalence). Therefore, absolute values are a common language to describe both the real embedding of
1191:
7143:
6930:
6010:
5756:
4503:
3209:
admits no real embeddings but admits a conjugate pair of complex embeddings. One of these embeddings sends
2713:
627:
494:
6925:
6910:
6846:
6819:
6589:
6243:
5772:
5726:
5231:
5188:
have six units. The integers in real quadratic number fields have infinitely many units. For example, in
3635:
2022:
1649:
659:
344:
4189:
3934:
6685:
5010:
1207:
104:
5777:
Dirichlet's unit theorem provides a description of the structure of the multiplicative group of units
804:
465:
428:
119:
7105:
7064:
6943:
5651:
3768:
3744:
2054:, and where this expression is unique up to the order of the factors. In particular, this is true if
1269:
1195:
1071:
1051:
5306:
3888:
1194:" refers to a long line of more concrete number theoretic statements which it generalized, from the
6949:
5850:
3856:
3734:
3494:. Because the Minkowski embedding is defined by field homomorphisms, multiplication of elements of
1168:
909:
721:, can resolve questions of primary importance in number theory, like the existence of solutions to
698:
579:
382:
332:
5837:) denotes the number of real embeddings (respectively, pairs of conjugate non-real embeddings) of
4111:
181:
7169:
6892:
4470:
2716:. These are formal objects which represent possible factorizations of numbers. The divisor group
2104:
125:
84:
7199:
7133:
6953:
6902:
6839:
6115:
Algebraic number theory interacts with many other mathematical disciplines. It uses tools from
5617:
5569:
4366:
3715:
2304:{\displaystyle 2\mathbf {Z} =(1+i)\mathbf {Z} \cdot (1-i)\mathbf {Z} =((1+i)\mathbf {Z} )^{2};}
1285:
1239:
1058:. He published important contributions to Fermat's last theorem, for which he proved the cases
1040:
1000:
686:
548:
399:
350:
131:
7173:
5038:
4674:
3484:
The subspace of the codomain fixed by complex conjugation is a real vector space of dimension
1226:
observed a possible link between two apparently completely distinct, branches of mathematics,
1159:. The concepts were highly influential, and his own contribution lives on in the names of the
7110:
7039:
6150:
6081:
5573:
3920:
3691:
1187:
964:
960:
768:
4340:
4012:
1512:
is not the rational numbers, however, there is no analog of positivity. For example, in the
7100:
6935:
6779:
6740:
6703:
6666:
6523:
6347:
6094:
6017:
5748:
5634:
5627:
5595:
4647:
4538:
4313:
2634:
represent the same element of the ideal class group if and only if there exists an element
1289:
1238:(at the time known as the Taniyama–Shimura conjecture) states that every elliptic curve is
1149:
1047:
1016:
952:
722:
706:
272:
146:
7163:
6787:
6673:
6654:
6531:
6335:
5755:
with norm less than a fixed positive integer . The order of the class group is called the
4598:
4569:
4224:
8:
7069:
6978:
6973:
6967:
6959:
6920:
6124:
6116:
5599:
5180:
4719:
3969:
3835:
complex places. Because places encompass the primes, places are sometimes referred to as
3675:
2974:
2418:
1657:
1199:
1160:
1141:
1067:
889:
718:
710:
554:
362:
313:
258:
152:
138:
66:
34:
7159:
4432:
1691:
is a unit. These are the elements that cannot be factored any further. Every element in
7115:
7059:
6963:
6880:
6140:
6041:
4627:
4563:
4412:
4293:
4273:
4253:
4166:
4146:
4047:
2734:
2724:
1458:
1281:
1273:
1265:
1235:
1156:
926:
870:
702:
567:
53:
6747:
6497:
4044:
has many absolute values, or places, and each corresponds to a point on the curve. If
7029:
6765:
6726:
6689:
6630:
6603:
6576:
6509:
6370:
6317:
This work established Takagi as Japan's first mathematician of international stature.
6299:
6271:
6209:
5557:
5257:
and the imaginary quadratic fields. A more precise statement giving the structure of
5251:. Thus, for example, the only fields for which the rank of the free part is zero are
3230:
2393:
1513:
1254:
1152:
in the dissertation of a student means his name is further attached to a major area.
1112:
1024:
690:
608:
405:
170:
111:
4363:
measures the order of vanishing or the order of a pole of a fraction of polynomials
7034:
7019:
6783:
6677:
6650:
6568:
6527:
6331:
5613:
5609:
3999:
2956:
2422:
2401:
2120:
1082:
992:
917:
682:
674:
614:
600:
414:
356:
319:
92:
78:
2849:{\displaystyle (x)={\mathfrak {p}}_{1}^{e_{1}}\cdots {\mathfrak {p}}_{t}^{e_{t}}.}
7048:
7024:
6939:
6775:
6736:
6722:
6699:
6662:
6624:
6597:
6519:
6343:
6203:
6155:
6029:
5885:
5879:
4003:
3587:
3511:
2062:
1888:
itself, so neither of them are prime. As there is no sense in which the elements
1223:
1036:
376:
326:
164:
2009:{\displaystyle I={\mathfrak {p}}_{1}^{e_{1}}\cdots {\mathfrak {p}}_{t}^{e_{t}},}
7085:
7004:
6888:
5752:
5663:
3679:
3099:, cannot. Abstractly, such a specification corresponds to a field homomorphism
2978:
2513:
1293:
1227:
1203:
1164:
1104:
1020:
420:
6572:
6444:
6419:
5560:, that describes both the quotient by this lattice and the ideal class group.
3853:
is a valuation corresponding to an absolute value, then one frequently writes
3470:{\displaystyle M\colon K\to \mathbf {R} ^{r_{1}}\oplus \mathbf {R} ^{2r_{2}}.}
1140:(literally "report on numbers"). He also resolved a significant number-theory
7188:
7044:
6896:
6862:
6145:
5271:
5221:
3995:
3165:
2405:
1131:
1108:
670:
561:
457:
72:
5633:
The zeta function is related to the other invariants described above by the
5285:
The free part of the unit group can be studied using the infinite places of
3801:. The other type of place is specified using a real or complex embedding of
3394:{\displaystyle M\colon K\to \mathbf {R} ^{r_{1}}\oplus \mathbf {C} ^{r_{2}}}
1250:
7090:
7014:
6914:
6327:
6104:
6060:). Hilbert reformulated the reciprocity laws as saying that a product over
6033:
4713:
2676:
1314:
1261:
1243:
1231:
1219:
1172:
1123:, devised as part of Kummer's 1843 attempt to prove Fermat's Last Theorem.
1120:
1116:
1028:
984:
754:
such that their sum, and the sum of their squares, equal two given numbers
714:
694:
593:
368:
264:
2141:. However, when this ideal is extended to the Gaussian integers to obtain
6360:
5646:
3515:
2051:
1136:
913:
897:
573:
284:
158:
40:
5576:, is an analytic object which describes the behavior of prime ideals in
1103:
included supplements introducing the notion of an ideal, fundamental to
7095:
7054:
6906:
6710:
6593:
6551:
6481:
6077:
5602:. The trivial character corresponds to the Riemann zeta function. When
3915:
3261:, while the number of conjugate pairs of complex embeddings is denoted
2103:. This extension field is now known as the Hilbert class field. By the
1309:
An important property of the ring of integers is that it satisfies the
1183:
1032:
996:
743:
739:
338:
1313:, that every (positive) integer has a factorization into a product of
3088:, can be specified as subfields of the real numbers. Others, such as
654:
298:
203:
5035:. For the place at infinity, this corresponds to the function field
3025:
1134:
unified the field of algebraic number theory with his 1897 treatise
1091:("Lectures on Number Theory") about which it has been written that:
5801:
is the finite cyclic group consisting of all the roots of unity in
2970:
678:
292:
278:
5888:, the law of quadratic reciprocity for positive odd primes states
3797:-adic absolute values, it measures divisibility. These are called
1535:
are associate because the latter is the product of the former by
176:
60:
6831:
3678:. Consider, for example, the integers. In addition to the usual
2088:
is a prime ideal which cannot be generated by a single element.
1376:
or a prime number. However, it is strictly weaker. For example,
979:
was the starting point for the work of other nineteenth century
6994:
6802:
5866: − 1 whose torsion consists of the roots of unity in
3718:
states that these are all possible absolute value functions on
980:
4143:
correspond to the places at infinity. Then, the completion of
3914:
Considering all the places of the field together produces the
2606:
The principal fractional ideals, meaning the ones of the form
662:, one of the founding works of modern algebraic number theory.
2150:, it may or may not be prime. For example, the factorization
1296:, and other 20th-century techniques not available to Fermat.
1257:, a list of important conjectures needing proof or disproof.
956:
944:
6684:, Cambridge Studies in Advanced Mathematics, vol. 27,
6233:"The Life and Work of Gustav Lejeune Dirichlet (1805–1859)"
5421:
3564:{\displaystyle \langle x,y\rangle =\operatorname {Tr} (xy)}
2945:{\displaystyle \operatorname {div} x=\sum _{i=1}^{t}e_{i}.}
5474:
to a real vector space. It can be shown that the image of
2712:
The ideal class group has another description in terms of
1050:, a basic counting argument, in the proof of a theorem in
937:
One of the founding works of algebraic number theory, the
6482:"A Computational Introduction to Algebraic Number Theory"
6342:(2nd ed.), London: 9780950273426, pp. 266–279,
3331:
Considering all embeddings at once determines a function
1803:{\displaystyle 9=3^{2}=(2+{\sqrt {-5}})(2-{\sqrt {-5}}).}
5218:
fundamental theorem of finitely generated abelian groups
3733:
of an algebraic number field is an equivalence class of
2682:
The number of elements in the class group is called the
880:). Solutions to linear Diophantine equations, such as 26
6496:
6468:
6202:
Gauss, Carl
Friedrich; Waterhouse, William C. (2018) ,
6127:. Algebraic number theory is also used in the study of
6085:
original quadratic reciprocity law can be hard to see.
2113:
generates a principal ideal of the ring of integers of
5230:. This group is cyclic. The free part is described by
2596:{\displaystyle J^{-1}=(O:J)=\{x\in K:xJ\subseteq O\}.}
1906:
can be made equivalent, unique factorization fails in
967:
and adds important new results of his own. Before the
7009:
6999:
6391:"At Last, Shout of 'Eureka!' In Age-Old Math Mystery"
5897:
5684:
5626:, and it has a factorization in terms of irreducible
5486:
5300:
5087:
5041:
5013:
4751:
4722:
4677:
4650:
4630:
4601:
4572:
4541:
4506:
4473:
4435:
4415:
4369:
4343:
4316:
4296:
4276:
4256:
4227:
4192:
4169:
4149:
4114:
4070:
4050:
4015:
3972:
3937:
3891:
3859:
3771:
3747:
3638:
3524:
3407:
3337:
3195:, respectively. Dually, an imaginary quadratic field
2990:
2875:
2775:
2525:
2376:. This, together with the observation that the ideal
2171:
2025:
1941:
1847:, but it does not, because all elements divisible by
1733:
1550:
1389:
951:) is a textbook of number theory written in Latin by
807:
771:
728:
506:
468:
431:
211:
184:
6265:
5149:{\displaystyle \sum _{n=-k}^{\infty }a_{n}(1/t)^{n}}
4644:. The function field of the completion at the place
2437:
which is closed under multiplication by elements of
1461:, meaning a number with a multiplicative inverse in
1171:. Results were mostly proved by 1930, after work by
6076:), taking values in roots of unity, is equal to 1.
3926:
1043:, a fundamental result in algebraic number theory.
6107:to a special value of its Dedekind zeta function.
5994:
5717:
5544:
5480:is a lattice that spans the hyperplane defined by
5427:
5234:. This theorem says that rank of the free part is
5148:
5070:
5027:
4996:
4734:
4704:
4663:
4636:
4616:
4587:
4554:
4527:
4492:
4459:
4421:
4401:
4355:
4329:
4302:
4282:
4262:
4242:
4213:
4175:
4155:
4135:
4101:{\displaystyle {\hat {X}}\subset \mathbb {A} ^{n}}
4100:
4056:
4036:
3986:
3958:
3903:
3871:
3781:
3757:
3658:
3563:
3469:
3393:
3061:
2944:
2848:
2595:
2421:structure. This is done by generalizing ideals to
2303:
2042:
2008:
1915:
1802:
1724:has two factorizations into irreducible elements,
1631:
1440:
1304:
1242:, meaning that it can be associated with a unique
845:
792:
530:
483:
446:
238:
192:
6599:Algebraic Number Theory and Fermat's Last Theorem
6553:Algebraic Number Theory, A Computational Approach
5678:of the rationals, one obtains a finite extension
5674:, if it is non-Archimedean and lies over a prime
3518:on Minkowski space corresponds to the trace form
3246:Conventionally, the number of real embeddings of
7186:
6565:A classical introduction to modern number theory
5738:
5545:{\displaystyle x_{1}+\cdots +x_{r_{1}+r_{2}}=0.}
4163:at one of these points gives an analogue of the
4064:is the projective completion of an affine curve
2675:(with the last notation identifying it with the
1206:. Artin's result provided a partial solution to
1202:and Kummer to Hilbert's product formula for the
1031:. Based on his research of the structure of the
6649:
6439:This notation indicates the ring obtained from
6414:This notation indicates the ring obtained from
6201:
5666:. If the valuation is Archimedean, one obtains
6672:
5201:is a unit, and all these powers are distinct.
3839:. When this is done, finite places are called
3072:
2981:of abelian groups (written multiplicatively),
6847:
6757:Grundlehren der mathematischen Wissenschaften
6506:Grundlehren der Mathematischen Wissenschaften
6330:(2010) , "History of Class Field Theory", in
5216:, is a finitely generated abelian group. The
2973:is the ideal class group. In the language of
2425:. A fractional ideal is an additive subgroup
920:in 1637, famously in the margin of a copy of
635:
6755:
6562:
6500:; Schmidt, Alexander; Wingberg, Kay (2000),
6044:another prime, and gave a relation between (
4290:is the field of fractions of polynomials in
3805:and the standard absolute value function on
3741:. There are two types of places. There is a
3537:
3525:
2587:
2560:
1720:are irreducible. This means that the number
239:{\displaystyle 0=\mathbb {Z} /1\mathbb {Z} }
6588:
5766:
1833:were a prime element, then it would divide
1145:
7149:
7139:
6854:
6840:
6508:, vol. 323, Berlin: Springer-Verlag,
6226:
6224:
6123:instead of number rings is referred to as
5594:, Dedekind zeta functions are products of
3765:-adic absolute value for each prime ideal
2347:. It implies that for an odd prime number
1632:{\displaystyle 5=(1+2i)(1-2i)=(2+i)(2-i),}
1441:{\displaystyle 6=2\cdot 3=(-2)\cdot (-3).}
1119:. Ideals generalize Ernst Eduard Kummer's
642:
628:
6563:Ireland, Kenneth; Rosen, Michael (2013),
5021:
4515:
4221:then its function field is isomorphic to
4201:
4088:
3946:
1479:is also a prime element. Numbers such as
531:{\displaystyle \mathbb {Z} (p^{\infty })}
508:
471:
434:
232:
219:
186:
6746:
6721:, vol. 110 (2 ed.), New York:
6266:Kanemitsu, Shigeru; Chaohua Jia (2002),
6259:
6230:
1884:, but neither of these elements divides
1107:. (The word "Ring", introduced later by
901:, of which only a portion has survived.
873:, originally solved by the Babylonians (
653:
6661:(2nd ed.), London: 9780950273426,
6643:
6268:Number theoretic methods: future trends
6221:
6103:relates many important invariants of a
6088:
5718:{\displaystyle K_{w}/\mathbf {Q} _{p}:}
5598:, with there being one factor for each
2345:Fermat's theorem on sums of two squares
1932:, then there is always a factorization
1011:In a couple of papers in 1838 and 1839
14:
7187:
6622:
6388:
5456:is the absolute value associated with
701:. These properties, such as whether a
6835:
6616:
6549:
6543:
6469:Neukirch, Schmidt & Wingberg 2000
6359:
6353:
6326:
5759:, and is often denoted by the letter
4528:{\displaystyle 2\in \mathbb {A} ^{1}}
1360:, then it divides one of the factors
1218:Around 1955, Japanese mathematicians
6709:
6289:
5747:is finite. This is a consequence of
5612:, the Dedekind zeta function is the
5572:of a number field, analogous to the
5552:The covolume of this lattice is the
2481:are fractional ideals, then the set
2411:
99:Free product of associative algebras
6382:
5873:
5751:since there are only finitely many
5444:varies over the infinite places of
3911:to mean that it is a finite place.
3774:
3750:
3659:{\displaystyle {\sqrt {|\Delta |}}}
3580:under the Minkowski embedding is a
3510:corresponds to multiplication by a
2925:
2820:
2791:
2043:{\displaystyle {\mathfrak {p}}_{i}}
2029:
1980:
1951:
1155:He made a series of conjectures on
658:Title page of the first edition of
24:
6538:
5204:In general, the group of units of
5164:The integers have only two units,
5107:
5078:which are power series of the form
4953:
4270:is an indeterminant and the field
4214:{\displaystyle X=\mathbb {P} ^{1}}
3959:{\displaystyle k=\mathbb {F} _{q}}
3898:
3866:
3646:
3596:is a basis for this lattice, then
3229:, while the other sends it to its
729:History of algebraic number theory
520:
25:
7211:
6861:
6795:
6479:
6129:arithmetic hyperbolic 3-manifolds
5028:{\displaystyle k\in \mathbb {N} }
2727:generated by the prime ideals of
2487:of all products of an element in
1311:fundamental theorem of arithmetic
1056:Dirichlet's approximation theorem
587:Noncommutative algebraic geometry
7168:
7158:
7148:
7138:
7129:
7128:
6801:
6110:
5847:finitely generated abelian group
5733:
5702:
5563:
5330:
4595:minus the order of vanishing of
3927:Places at infinity geometrically
3710:, which measure divisibility by
3706:, defined for each prime number
3514:in the Minkowski embedding. The
3444:
3422:
3374:
3352:
2275:
2240:
2208:
2176:
1299:
1213:
846:{\displaystyle B=x^{2}+y^{2}.\ }
484:{\displaystyle \mathbb {Q} _{p}}
447:{\displaystyle \mathbb {Z} _{p}}
6473:
6467:See proposition VIII.8.6.11 of
6461:
6433:
6408:
5785:. Specifically, it states that
5640:
3843:and infinite places are called
3782:{\displaystyle {\mathfrak {p}}}
3758:{\displaystyle {\mathfrak {p}}}
3626:. The covolume of the image of
2469:are also fractional ideals. If
1916:Factorization into prime ideals
1305:Failure of unique factorization
895:Diophantus' major work was the
6907:analytic theory of L-functions
6885:non-abelian class field theory
6320:
6311:
6283:
6195:
6186:
6177:
6168:
5947:
5937:
5409:
5398:
5389:
5379:
5373:
5367:
5325:
5137:
5122:
5065:
5062:
5048:
5045:
4981:
4968:
4909:
4896:
4880:
4868:
4830:
4817:
4783:
4770:
4742:, so an element is of the form
4699:
4696:
4684:
4681:
4611:
4605:
4582:
4576:
4500:this corresponds to the point
4454:
4442:
4396:
4390:
4379:
4373:
4237:
4231:
4127:
4077:
4031:
4025:
3904:{\displaystyle v\nmid \infty }
3650:
3642:
3616:. The discriminant is denoted
3558:
3549:
3417:
3347:
3053:
3041:
3007:
2994:
2936:
2919:
2782:
2776:
2554:
2542:
2289:
2285:
2279:
2271:
2259:
2256:
2250:
2244:
2236:
2224:
2218:
2212:
2204:
2192:
2186:
2180:
1794:
1775:
1772:
1753:
1623:
1611:
1608:
1596:
1590:
1575:
1572:
1557:
1495:. In the integers, the primes
1432:
1423:
1417:
1408:
1099:1879 and 1894 editions of the
1088:Vorlesungen über Zahlentheorie
1023:(later refined by his student
1013:Peter Gustav Lejeune Dirichlet
989:Peter Gustav Lejeune Dirichlet
525:
512:
13:
1:
6719:Graduate Texts in Mathematics
6389:Kolata, Gina (24 June 1993).
5739:Finiteness of the class group
3872:{\displaystyle v\mid \infty }
2977:, this says that there is an
2866:is defined to be the divisor
2135:are prime ideals of the ring
1323:of an algebraic number field
888:= 13, may be found using the
874:
733:
6931:Transcendental number theory
6270:, Springer, pp. 271–4,
6240:Clay Mathematics Proceedings
6030:quadratic reciprocity symbol
6011:law of quadratic reciprocity
4136:{\displaystyle X-{\hat {X}}}
3077:Some number fields, such as
2369:and is not a prime ideal if
1702:. In this ring, the numbers
1650:unique factorization domains
1198:and the reciprocity laws of
1142:problem formulated by Waring
1066: = 14, and to the
1006:
673:that uses the techniques of
193:{\displaystyle \mathbb {Z} }
7:
7154:List of recreational topics
6926:Computational number theory
6911:probabilistic number theory
6820:Encyclopedia of Mathematics
6502:Cohomology of Number Fields
6205:Disquisitiones Arithmeticae
6134:
6009:is a generalization of the
5588:is an abelian extension of
4493:{\displaystyle x_{1}\neq 0}
3073:Real and complex embeddings
2743:, the non-zero elements of
1669:is an element such that if
1077:
1068:biquadratic reciprocity law
949:Arithmetical Investigations
940:Disquisitiones Arithmeticae
660:Disquisitiones Arithmeticae
345:Unique factorization domain
10:
7216:
6752:Algebraische Zahlentheorie
6686:Cambridge University Press
6629:(2nd ed.), Springer,
6623:Marcus, Daniel A. (2018),
6567:, vol. 84, Springer,
6092:
5877:
5770:
5644:
2329:, the ideals generated by
2080:, for instance, the ideal
1270:semistable elliptic curves
1126:
105:Tensor product of algebras
7124:
7106:Diophantine approximation
7078:
7065:Chinese remainder theorem
6987:
6869:
6815:"Algebraic number theory"
6760:. Vol. 322. Berlin:
6573:10.1007/978-1-4757-2103-4
6231:Elstrodt, Jürgen (2007),
4467:, so on the affine chart
4402:{\displaystyle p(x)/q(x)}
3885:is an infinite place and
3669:
2963:is the group of units in
2749:up to multiplication, to
1813:This equation shows that
1473:is a prime element, then
1196:quadratic reciprocity law
1072:Dirichlet divisor problem
1052:diophantine approximation
983:mathematicians including
904:
6950:Arithmetic combinatorics
6290:Reid, Constance (1996),
6161:
6032:, that describes when a
5781:of the ring of integers
5773:Dirichlet's unit theorem
5767:Dirichlet's unit theorem
5291:. Consider the function
5274:for the Galois group of
5232:Dirichlet's unit theorem
5159:
5071:{\displaystyle k((1/t))}
4705:{\displaystyle k((t-2))}
2679:in algebraic geometry).
1264:provided a proof of the
1178:
1169:local class field theory
1054:, later named after him
932:
383:Formal power series ring
333:Integrally closed domain
7195:Algebraic number theory
6921:Geometric number theory
6877:Algebraic number theory
6808:Algebraic number theory
6715:Algebraic number theory
6682:Algebraic number theory
6659:Algebraic number theory
6550:Stein, William (2012),
6340:Algebraic number theory
5727:Kronecker–Weber theorem
5620:of the Galois group of
5468:is a homomorphism from
3726:and the prime numbers.
3296:. It is a theorem that
3130:A real quadratic field
2107:, every prime ideal of
2105:principal ideal theorem
1272:, which, together with
1208:Hilbert's ninth problem
793:{\displaystyle A=x+y\ }
687:algebraic number fields
667:Algebraic number theory
392:Algebraic number theory
85:Total ring of fractions
27:Branch of number theory
7040:Transcendental numbers
6954:additive number theory
6903:Analytic number theory
6756:
5996:
5823: − 1 (where
5719:
5618:regular representation
5570:Dedekind zeta function
5546:
5429:
5157:
5150:
5111:
5072:
5029:
5005:
4998:
4957:
4736:
4712:which is the field of
4706:
4665:
4638:
4618:
4589:
4556:
4529:
4494:
4461:
4423:
4403:
4357:
4356:{\displaystyle p\in X}
4331:
4304:
4284:
4264:
4244:
4215:
4177:
4157:
4137:
4102:
4058:
4038:
4037:{\displaystyle F=k(X)}
3988:
3960:
3905:
3873:
3783:
3759:
3660:
3565:
3471:
3395:
3063:
2946:
2908:
2850:
2690:. The class number of
2597:
2305:
2044:
2010:
1804:
1633:
1442:
1146:the finiteness theorem
1097:
1041:Dirichlet unit theorem
1001:complex multiplication
847:
794:
663:
549:Noncommutative algebra
532:
485:
448:
400:Algebraic number field
351:Principal ideal domain
240:
194:
132:Frobenius endomorphism
7111:Irrationality measure
7101:Diophantine equations
6944:Hodge–Arakelov theory
6366:Fermat's Last Theorem
6151:Locally compact field
5997:
5720:
5630:of the Galois group.
5628:Artin representations
5596:Dirichlet L-functions
5574:Riemann zeta function
5547:
5430:
5151:
5088:
5080:
5073:
5030:
4999:
4934:
4744:
4737:
4707:
4666:
4664:{\displaystyle v_{2}}
4639:
4619:
4590:
4557:
4555:{\displaystyle v_{2}}
4530:
4495:
4462:
4424:
4404:
4358:
4332:
4330:{\displaystyle v_{p}}
4305:
4285:
4265:
4245:
4216:
4178:
4158:
4138:
4103:
4059:
4039:
3989:
3961:
3921:Artin reciprocity law
3906:
3874:
3784:
3760:
3692:p-adic absolute value
3661:
3566:
3472:
3396:
3064:
2947:
2888:
2851:
2723:is defined to be the
2598:
2506:, and the inverse of
2306:
2045:
2011:
1805:
1656:prime element and an
1634:
1443:
1188:Artin reciprocity law
1150:Hilbert modular forms
1093:
929:in the 20th century.
910:Fermat's Last Theorem
892:(c. 5th century BC).
848:
795:
723:Diophantine equations
657:
533:
486:
449:
241:
195:
7070:Arithmetic functions
6936:Diophantine geometry
6810:at Wikimedia Commons
6644:Graduate level texts
6101:class number formula
6095:Class number formula
6089:Class number formula
6064:of Hilbert symbols (
6018:power residue symbol
5895:
5682:
5635:class number formula
5484:
5298:
5085:
5039:
5011:
4749:
4720:
4675:
4648:
4628:
4617:{\displaystyle q(x)}
4599:
4588:{\displaystyle p(x)}
4570:
4539:
4504:
4471:
4433:
4413:
4367:
4341:
4314:
4294:
4274:
4254:
4243:{\displaystyle k(t)}
4225:
4190:
4167:
4147:
4112:
4068:
4048:
4013:
3970:
3935:
3889:
3857:
3769:
3745:
3682:function |·| :
3636:
3522:
3405:
3335:
2988:
2873:
2773:
2523:
2362:is a prime ideal if
2169:
2023:
1939:
1817:divides the product
1731:
1642:which prove that in
1548:
1387:
1048:pigeonhole principle
1017:class number formula
953:Carl Friedrich Gauss
805:
769:
707:unique factorization
555:Noncommutative rings
504:
466:
429:
273:Non-associative ring
209:
182:
139:Algebraic structures
7116:Continued fractions
6979:Arithmetic dynamics
6974:Arithmetic topology
6968:P-adic Hodge theory
6960:Arithmetic geometry
6893:Iwasawa–Tate theory
6174:Stark, pp. 145–146.
6125:arithmetic geometry
6117:homological algebra
6028:) generalizing the
5841:). In other words,
5749:Minkowski's theorem
5600:Dirichlet character
5181:Eisenstein integers
4735:{\displaystyle t-2}
4108:then the points in
3987:{\displaystyle X/k}
3716:Ostrowski's theorem
3479:Minkowski embedding
3477:This is called the
3119:. These are called
3029:
2975:homological algebra
2842:
2813:
2512:is a (generalized)
2002:
1973:
1880:divide the product
1662:irreducible element
1658:irreducible element
1260:From 1993 to 1994,
1161:Hilbert class field
1062: = 5 and
890:Euclidean algorithm
871:Pythagorean triples
314:Commutative algebra
153:Associative algebra
35:Algebraic structure
18:Place (mathematics)
7060:Modular arithmetic
7030:Irrational numbers
6964:anabelian geometry
6881:class field theory
6674:Fröhlich, Albrecht
6655:Fröhlich, Albrecht
6617:Intermediate texts
6544:Introductory texts
6395:The New York Times
6336:Fröhlich, Albrecht
6141:Class field theory
5992:
5715:
5542:
5425:
5420:
5282:is also possible.
5146:
5068:
5025:
4994:
4992:
4732:
4702:
4661:
4634:
4614:
4585:
4564:order of vanishing
4552:
4525:
4490:
4460:{\displaystyle p=}
4457:
4429:. For example, if
4419:
4399:
4353:
4327:
4300:
4280:
4260:
4240:
4211:
4173:
4153:
4133:
4098:
4054:
4034:
3984:
3956:
3901:
3869:
3779:
3755:
3656:
3561:
3467:
3401:, or equivalently
3391:
3125:complex embeddings
3059:
2942:
2846:
2817:
2788:
2735:group homomorphism
2725:free abelian group
2593:
2493:and an element in
2417:that they admit a
2314:note that because
2301:
2040:
2006:
1977:
1948:
1800:
1629:
1438:
1354:divides a product
1282:algebraic geometry
1266:modularity theorem
1236:modularity theorem
1157:class field theory
1113:algebraic integers
1046:He first used the
927:modularity theorem
843:
790:
709:, the behavior of
664:
568:Semiprimitive ring
528:
481:
444:
252:Related structures
236:
190:
126:Inner automorphism
112:Ring homomorphisms
7182:
7181:
7079:Advanced concepts
7035:Algebraic numbers
7020:Composite numbers
6806:Media related to
6771:978-3-540-65399-8
6732:978-0-387-94225-4
6678:Taylor, Martin J.
6651:Cassels, J. W. S.
6636:978-3-319-90233-3
6609:978-1-4987-3840-8
6582:978-1-4757-2103-4
6515:978-3-540-66671-4
6369:, Fourth Estate,
6332:Cassels, J. W. S.
6277:978-1-4020-1080-4
6215:978-1-4939-7560-0
6192:Stark, pp. 44–47.
6183:Aczel, pp. 14–15.
6040:th power residue
5985:
5967:
5928:
5910:
5789:is isomorphic to
5558:idele class group
5194:, every power of
4637:{\displaystyle 2}
4422:{\displaystyle p}
4303:{\displaystyle t}
4283:{\displaystyle F}
4263:{\displaystyle t}
4176:{\displaystyle p}
4156:{\displaystyle F}
4130:
4080:
4057:{\displaystyle X}
3654:
3322:is the degree of
3231:complex conjugate
3030:
3028:
2423:fractional ideals
2412:Ideal class group
2394:abelian extension
2121:cyclotomic fields
1792:
1770:
1514:Gaussian integers
1255:Langlands program
1144:in 1770. As with
1025:Leopold Kronecker
1015:proved the first
1003:, in particular.
925:the proof of the
869:are given by the
842:
789:
691:rings of integers
652:
651:
609:Geometric algebra
320:Commutative rings
171:Category of rings
16:(Redirected from
7207:
7172:
7162:
7152:
7151:
7142:
7141:
7132:
7131:
7025:Rational numbers
6856:
6849:
6842:
6833:
6832:
6828:
6805:
6791:
6759:
6748:Neukirch, Jürgen
6743:
6706:
6669:
6657:, eds. (2010) ,
6639:
6612:
6585:
6559:
6558:
6534:
6498:Neukirch, Jürgen
6489:
6488:
6486:
6477:
6471:
6465:
6459:
6457:
6456:
6437:
6431:
6412:
6406:
6405:
6403:
6401:
6386:
6380:
6379:
6357:
6351:
6350:
6324:
6318:
6315:
6309:
6308:
6287:
6281:
6280:
6263:
6257:
6256:
6255:
6254:
6248:
6242:, archived from
6237:
6228:
6219:
6218:
6199:
6193:
6190:
6184:
6181:
6175:
6172:
6080:'s reformulated
6001:
5999:
5998:
5993:
5988:
5987:
5986:
5981:
5970:
5968:
5963:
5952:
5933:
5929:
5921:
5915:
5911:
5903:
5884:In terms of the
5874:Reciprocity laws
5724:
5722:
5721:
5716:
5711:
5710:
5705:
5699:
5694:
5693:
5625:
5614:Artin L-function
5610:Galois extension
5607:
5593:
5587:
5581:
5551:
5549:
5548:
5543:
5535:
5534:
5533:
5532:
5520:
5519:
5496:
5495:
5479:
5473:
5467:
5461:
5449:
5443:
5434:
5432:
5431:
5426:
5424:
5423:
5417:
5416:
5407:
5406:
5401:
5392:
5359:
5358:
5357:
5356:
5344:
5343:
5333:
5324:
5323:
5290:
5256:
5250:
5229:
5215:
5209:
5200:
5199:
5193:
5187:
5178:
5171:
5167:
5155:
5153:
5152:
5147:
5145:
5144:
5132:
5121:
5120:
5110:
5105:
5077:
5075:
5074:
5069:
5058:
5034:
5032:
5031:
5026:
5024:
5003:
5001:
5000:
4995:
4993:
4989:
4988:
4967:
4966:
4956:
4951:
4927:
4917:
4916:
4895:
4894:
4867:
4866:
4854:
4853:
4841:
4840:
4816:
4815:
4794:
4793:
4769:
4768:
4755:
4741:
4739:
4738:
4733:
4716:in the variable
4711:
4709:
4708:
4703:
4670:
4668:
4667:
4662:
4660:
4659:
4643:
4641:
4640:
4635:
4623:
4621:
4620:
4615:
4594:
4592:
4591:
4586:
4561:
4559:
4558:
4553:
4551:
4550:
4535:, the valuation
4534:
4532:
4531:
4526:
4524:
4523:
4518:
4499:
4497:
4496:
4491:
4483:
4482:
4466:
4464:
4463:
4458:
4428:
4426:
4425:
4420:
4408:
4406:
4405:
4400:
4386:
4362:
4360:
4359:
4354:
4336:
4334:
4333:
4328:
4326:
4325:
4310:. Then, a place
4309:
4307:
4306:
4301:
4289:
4287:
4286:
4281:
4269:
4267:
4266:
4261:
4249:
4247:
4246:
4241:
4220:
4218:
4217:
4212:
4210:
4209:
4204:
4186:For example, if
4182:
4180:
4179:
4174:
4162:
4160:
4159:
4154:
4142:
4140:
4139:
4134:
4132:
4131:
4123:
4107:
4105:
4104:
4099:
4097:
4096:
4091:
4082:
4081:
4073:
4063:
4061:
4060:
4055:
4043:
4041:
4040:
4035:
3993:
3991:
3990:
3985:
3980:
3965:
3963:
3962:
3957:
3955:
3954:
3949:
3910:
3908:
3907:
3902:
3884:
3878:
3876:
3875:
3870:
3852:
3834:
3826:real places and
3825:
3793:, and, like the
3788:
3786:
3785:
3780:
3778:
3777:
3764:
3762:
3761:
3756:
3754:
3753:
3665:
3663:
3662:
3657:
3655:
3653:
3645:
3640:
3631:
3625:
3619:
3615:
3605:
3595:
3585:
3579:
3570:
3568:
3567:
3562:
3509:
3499:
3489:
3476:
3474:
3473:
3468:
3463:
3462:
3461:
3460:
3447:
3438:
3437:
3436:
3435:
3425:
3400:
3398:
3397:
3392:
3390:
3389:
3388:
3387:
3377:
3368:
3367:
3366:
3365:
3355:
3327:
3321:
3315:
3295:
3269:
3260:
3251:
3242:
3241:
3228:
3227:
3218:
3217:
3208:
3206:
3194:
3193:
3185:
3184:
3176:
3175:
3163:
3157:
3142:
3140:
3127:, respectively.
3118:
3108:
3098:
3096:
3087:
3085:
3068:
3066:
3065:
3060:
3031:
3026:
3021:
3019:
3018:
3006:
3005:
2968:
2962:
2951:
2949:
2948:
2943:
2935:
2934:
2929:
2928:
2918:
2917:
2907:
2902:
2865:
2855:
2853:
2852:
2847:
2841:
2840:
2839:
2829:
2824:
2823:
2812:
2811:
2810:
2800:
2795:
2794:
2765:
2755:
2748:
2742:
2732:
2722:
2708:
2706:
2700:
2698:
2674:
2667:
2660:
2653:
2643:
2633:
2627:
2621:
2611:
2602:
2600:
2599:
2594:
2538:
2537:
2511:
2505:
2498:
2492:
2486:
2480:
2474:
2468:
2463:. All ideals of
2462:
2452:
2442:
2436:
2430:
2402:Galois extension
2386:
2375:
2368:
2361:
2352:
2342:
2335:
2328:
2310:
2308:
2307:
2302:
2297:
2296:
2278:
2243:
2211:
2179:
2161:
2149:
2140:
2134:
2118:
2112:
2102:
2096:
2087:
2085:
2079:
2073:
2063:Dedekind domains
2059:
2049:
2047:
2046:
2041:
2039:
2038:
2033:
2032:
2015:
2013:
2012:
2007:
2001:
2000:
1999:
1989:
1984:
1983:
1972:
1971:
1970:
1960:
1955:
1954:
1931:
1925:
1911:
1905:
1904:
1898:
1897:
1891:
1887:
1883:
1879:
1878:
1872:
1871:
1865:
1864:
1851:are of the form
1850:
1846:
1845:
1839:
1838:
1832:
1828:
1826:
1822:
1816:
1809:
1807:
1806:
1801:
1793:
1785:
1771:
1763:
1749:
1748:
1723:
1719:
1718:
1712:
1711:
1705:
1701:
1690:
1684:
1678:
1668:
1647:
1638:
1636:
1635:
1630:
1540:
1534:
1527:
1520:
1507:
1500:
1490:
1484:
1478:
1472:
1466:
1456:
1447:
1445:
1444:
1439:
1379:
1375:
1371:
1365:
1359:
1353:
1347:
1341:
1328:
1322:
1234:. The resulting
1186:established the
1083:Richard Dedekind
1039:, he proved the
1037:quadratic fields
993:Richard Dedekind
918:Pierre de Fermat
879:
876:
852:
850:
849:
844:
840:
836:
835:
823:
822:
799:
797:
796:
791:
787:
762:, respectively:
683:rational numbers
675:abstract algebra
644:
637:
630:
615:Operator algebra
601:Clifford algebra
537:
535:
534:
529:
524:
523:
511:
490:
488:
487:
482:
480:
479:
474:
453:
451:
450:
445:
443:
442:
437:
415:Ring of integers
409:
406:Integers modulo
357:Euclidean domain
245:
243:
242:
237:
235:
227:
222:
199:
197:
196:
191:
189:
93:Product of rings
79:Fractional ideal
38:
30:
29:
21:
7215:
7214:
7210:
7209:
7208:
7206:
7205:
7204:
7185:
7184:
7183:
7178:
7120:
7086:Quadratic forms
7074:
7049:P-adic analysis
7005:Natural numbers
6983:
6940:Arakelov theory
6865:
6860:
6813:
6798:
6772:
6762:Springer-Verlag
6733:
6723:Springer-Verlag
6696:
6646:
6637:
6619:
6610:
6583:
6556:
6546:
6541:
6539:Further reading
6516:
6493:
6492:
6484:
6478:
6474:
6466:
6462:
6454:
6452:
6438:
6434:
6413:
6409:
6399:
6397:
6387:
6383:
6377:
6358:
6354:
6325:
6321:
6316:
6312:
6306:
6288:
6284:
6278:
6264:
6260:
6252:
6250:
6246:
6235:
6229:
6222:
6216:
6200:
6196:
6191:
6187:
6182:
6178:
6173:
6169:
6164:
6156:Tamagawa number
6137:
6113:
6097:
6091:
6082:reciprocity law
6007:reciprocity law
5971:
5969:
5953:
5951:
5950:
5946:
5920:
5916:
5902:
5898:
5896:
5893:
5892:
5886:Legendre symbol
5882:
5880:Reciprocity law
5876:
5865:
5858:
5836:
5830:(respectively,
5829:
5822:
5815:
5775:
5769:
5753:Integral ideals
5741:
5736:
5706:
5701:
5700:
5695:
5689:
5685:
5683:
5680:
5679:
5654:a number field
5649:
5643:
5621:
5603:
5589:
5583:
5577:
5566:
5528:
5524:
5515:
5511:
5510:
5506:
5491:
5487:
5485:
5482:
5481:
5475:
5469:
5463:
5462:. The function
5457:
5455:
5445:
5439:
5419:
5418:
5412:
5408:
5402:
5397:
5396:
5388:
5361:
5360:
5352:
5348:
5339:
5335:
5334:
5329:
5328:
5319:
5315:
5302:
5301:
5299:
5296:
5295:
5286:
5266:
5252:
5248:
5241:
5235:
5225:
5211:
5205:
5197:
5195:
5189:
5183:
5173:
5169:
5165:
5162:
5140:
5136:
5128:
5116:
5112:
5106:
5092:
5086:
5083:
5082:
5054:
5040:
5037:
5036:
5020:
5012:
5009:
5008:
4991:
4990:
4984:
4980:
4962:
4958:
4952:
4938:
4925:
4924:
4912:
4908:
4890:
4886:
4862:
4858:
4849:
4845:
4833:
4829:
4808:
4804:
4786:
4782:
4761:
4757:
4752:
4750:
4747:
4746:
4721:
4718:
4717:
4676:
4673:
4672:
4655:
4651:
4649:
4646:
4645:
4629:
4626:
4625:
4600:
4597:
4596:
4571:
4568:
4567:
4546:
4542:
4540:
4537:
4536:
4519:
4514:
4513:
4505:
4502:
4501:
4478:
4474:
4472:
4469:
4468:
4434:
4431:
4430:
4414:
4411:
4410:
4382:
4368:
4365:
4364:
4342:
4339:
4338:
4321:
4317:
4315:
4312:
4311:
4295:
4292:
4291:
4275:
4272:
4271:
4255:
4252:
4251:
4226:
4223:
4222:
4205:
4200:
4199:
4191:
4188:
4187:
4168:
4165:
4164:
4148:
4145:
4144:
4122:
4121:
4113:
4110:
4109:
4092:
4087:
4086:
4072:
4071:
4069:
4066:
4065:
4049:
4046:
4045:
4014:
4011:
4010:
4004:algebraic curve
3976:
3971:
3968:
3967:
3950:
3945:
3944:
3936:
3933:
3932:
3929:
3890:
3887:
3886:
3880:
3858:
3855:
3854:
3848:
3845:infinite primes
3833:
3827:
3824:
3818:
3815:infinite places
3773:
3772:
3770:
3767:
3766:
3749:
3748:
3746:
3743:
3742:
3697:
3672:
3649:
3641:
3639:
3637:
3634:
3633:
3627:
3621:
3617:
3611:
3597:
3591:
3581:
3575:
3523:
3520:
3519:
3512:diagonal matrix
3501:
3495:
3492:Minkowski space
3485:
3456:
3452:
3448:
3443:
3442:
3431:
3427:
3426:
3421:
3420:
3406:
3403:
3402:
3383:
3379:
3378:
3373:
3372:
3361:
3357:
3356:
3351:
3350:
3336:
3333:
3332:
3323:
3317:
3310:
3303:
3297:
3293:
3286:
3279:
3268:
3262:
3259:
3253:
3247:
3236:
3234:
3222:
3220:
3212:
3210:
3201:
3196:
3189:
3187:
3180:
3178:
3171:
3169:
3159:
3144:
3136:
3131:
3121:real embeddings
3110:
3100:
3094:
3089:
3083:
3078:
3075:
3020:
3014:
3010:
3001:
2997:
2989:
2986:
2985:
2964:
2960:
2930:
2924:
2923:
2922:
2913:
2909:
2903:
2892:
2874:
2871:
2870:
2860:
2835:
2831:
2830:
2825:
2819:
2818:
2806:
2802:
2801:
2796:
2790:
2789:
2774:
2771:
2770:
2757:
2756:. Suppose that
2750:
2744:
2738:
2728:
2717:
2704:
2702:
2696:
2691:
2669:
2662:
2655:
2645:
2635:
2629:
2623:
2613:
2607:
2530:
2526:
2524:
2521:
2520:
2507:
2500:
2494:
2488:
2482:
2476:
2470:
2464:
2454:
2444:
2443:, meaning that
2438:
2432:
2426:
2414:
2408:Galois group).
2377:
2370:
2363:
2354:
2348:
2337:
2330:
2315:
2292:
2288:
2274:
2239:
2207:
2175:
2170:
2167:
2166:
2151:
2142:
2136:
2127:
2114:
2108:
2098:
2092:
2083:
2081:
2075:
2069:
2055:
2034:
2028:
2027:
2026:
2024:
2021:
2020:
1995:
1991:
1990:
1985:
1979:
1978:
1966:
1962:
1961:
1956:
1950:
1949:
1940:
1937:
1936:
1927:
1926:is an ideal in
1921:
1918:
1907:
1902:
1900:
1895:
1893:
1889:
1885:
1881:
1876:
1874:
1869:
1867:
1862:
1852:
1848:
1843:
1841:
1836:
1834:
1830:
1824:
1820:
1818:
1814:
1784:
1762:
1744:
1740:
1732:
1729:
1728:
1721:
1716:
1714:
1709:
1707:
1703:
1697:
1686:
1680:
1670:
1664:
1643:
1549:
1546:
1545:
1536:
1529:
1522:
1516:
1502:
1496:
1491:are said to be
1486:
1480:
1474:
1468:
1462:
1452:
1451:In general, if
1388:
1385:
1384:
1377:
1373:
1367:
1361:
1355:
1349:
1343:
1337:
1324:
1318:
1307:
1302:
1274:Ribet's theorem
1228:elliptic curves
1224:Yutaka Taniyama
1216:
1192:reciprocity law
1181:
1129:
1080:
1021:quadratic forms
1009:
935:
907:
877:
857:
831:
827:
818:
814:
806:
803:
802:
770:
767:
766:
742:mathematician,
736:
731:
699:function fields
669:is a branch of
648:
619:
618:
551:
541:
540:
519:
515:
507:
505:
502:
501:
475:
470:
469:
467:
464:
463:
438:
433:
432:
430:
427:
426:
407:
377:Polynomial ring
327:Integral domain
316:
306:
305:
231:
223:
218:
210:
207:
206:
185:
183:
180:
179:
165:Involutive ring
50:
39:
33:
28:
23:
22:
15:
12:
11:
5:
7213:
7203:
7202:
7197:
7180:
7179:
7177:
7176:
7166:
7156:
7146:
7144:List of topics
7136:
7125:
7122:
7121:
7119:
7118:
7113:
7108:
7103:
7098:
7093:
7088:
7082:
7080:
7076:
7075:
7073:
7072:
7067:
7062:
7057:
7052:
7045:P-adic numbers
7042:
7037:
7032:
7027:
7022:
7017:
7012:
7007:
7002:
6997:
6991:
6989:
6985:
6984:
6982:
6981:
6976:
6971:
6957:
6947:
6933:
6928:
6923:
6918:
6900:
6889:Iwasawa theory
6873:
6871:
6867:
6866:
6859:
6858:
6851:
6844:
6836:
6830:
6829:
6811:
6797:
6796:External links
6794:
6793:
6792:
6770:
6744:
6731:
6707:
6694:
6670:
6645:
6642:
6641:
6640:
6635:
6618:
6615:
6614:
6613:
6608:
6586:
6581:
6560:
6545:
6542:
6540:
6537:
6536:
6535:
6514:
6491:
6490:
6472:
6460:
6432:
6407:
6381:
6375:
6352:
6319:
6310:
6304:
6282:
6276:
6258:
6220:
6214:
6194:
6185:
6176:
6166:
6165:
6163:
6160:
6159:
6158:
6153:
6148:
6143:
6136:
6133:
6112:
6109:
6093:Main article:
6090:
6087:
6003:
6002:
5991:
5984:
5980:
5977:
5974:
5966:
5962:
5959:
5956:
5949:
5945:
5942:
5939:
5936:
5932:
5927:
5924:
5919:
5914:
5909:
5906:
5901:
5878:Main article:
5875:
5872:
5863:
5856:
5834:
5827:
5820:
5813:
5771:Main article:
5768:
5765:
5740:
5737:
5735:
5732:
5714:
5709:
5704:
5698:
5692:
5688:
5664:complete field
5645:Main article:
5642:
5639:
5565:
5562:
5541:
5538:
5531:
5527:
5523:
5518:
5514:
5509:
5505:
5502:
5499:
5494:
5490:
5451:
5436:
5435:
5422:
5415:
5411:
5405:
5400:
5395:
5391:
5387:
5384:
5381:
5378:
5375:
5372:
5369:
5366:
5363:
5362:
5355:
5351:
5347:
5342:
5338:
5332:
5327:
5322:
5318:
5314:
5311:
5308:
5307:
5305:
5262:
5246:
5239:
5222:roots of unity
5161:
5158:
5143:
5139:
5135:
5131:
5127:
5124:
5119:
5115:
5109:
5104:
5101:
5098:
5095:
5091:
5067:
5064:
5061:
5057:
5053:
5050:
5047:
5044:
5023:
5019:
5016:
4987:
4983:
4979:
4976:
4973:
4970:
4965:
4961:
4955:
4950:
4947:
4944:
4941:
4937:
4933:
4930:
4928:
4926:
4923:
4920:
4915:
4911:
4907:
4904:
4901:
4898:
4893:
4889:
4885:
4882:
4879:
4876:
4873:
4870:
4865:
4861:
4857:
4852:
4848:
4844:
4839:
4836:
4832:
4828:
4825:
4822:
4819:
4814:
4811:
4807:
4803:
4800:
4797:
4792:
4789:
4785:
4781:
4778:
4775:
4772:
4767:
4764:
4760:
4756:
4754:
4731:
4728:
4725:
4701:
4698:
4695:
4692:
4689:
4686:
4683:
4680:
4658:
4654:
4633:
4613:
4610:
4607:
4604:
4584:
4581:
4578:
4575:
4549:
4545:
4522:
4517:
4512:
4509:
4489:
4486:
4481:
4477:
4456:
4453:
4450:
4447:
4444:
4441:
4438:
4418:
4398:
4395:
4392:
4389:
4385:
4381:
4378:
4375:
4372:
4352:
4349:
4346:
4324:
4320:
4299:
4279:
4259:
4239:
4236:
4233:
4230:
4208:
4203:
4198:
4195:
4172:
4152:
4129:
4126:
4120:
4117:
4095:
4090:
4085:
4079:
4076:
4053:
4033:
4030:
4027:
4024:
4021:
4018:
4008:function field
3983:
3979:
3975:
3953:
3948:
3943:
3940:
3928:
3925:
3900:
3897:
3894:
3868:
3865:
3862:
3831:
3822:
3776:
3752:
3735:absolute value
3695:
3680:absolute value
3671:
3668:
3652:
3648:
3644:
3560:
3557:
3554:
3551:
3548:
3545:
3542:
3539:
3536:
3533:
3530:
3527:
3500:by an element
3466:
3459:
3455:
3451:
3446:
3441:
3434:
3430:
3424:
3419:
3416:
3413:
3410:
3386:
3382:
3376:
3371:
3364:
3360:
3354:
3349:
3346:
3343:
3340:
3308:
3301:
3291:
3284:
3266:
3257:
3166:perfect square
3074:
3071:
3070:
3069:
3058:
3055:
3052:
3049:
3046:
3043:
3040:
3037:
3034:
3024:
3017:
3013:
3009:
3004:
3000:
2996:
2993:
2979:exact sequence
2953:
2952:
2941:
2938:
2933:
2927:
2921:
2916:
2912:
2906:
2901:
2898:
2895:
2891:
2887:
2884:
2881:
2878:
2857:
2856:
2845:
2838:
2834:
2828:
2822:
2816:
2809:
2805:
2799:
2793:
2787:
2784:
2781:
2778:
2604:
2603:
2592:
2589:
2586:
2583:
2580:
2577:
2574:
2571:
2568:
2565:
2562:
2559:
2556:
2553:
2550:
2547:
2544:
2541:
2536:
2533:
2529:
2514:ideal quotient
2413:
2410:
2312:
2311:
2300:
2295:
2291:
2287:
2284:
2281:
2277:
2273:
2270:
2267:
2264:
2261:
2258:
2255:
2252:
2249:
2246:
2242:
2238:
2235:
2232:
2229:
2226:
2223:
2220:
2217:
2214:
2210:
2206:
2203:
2200:
2197:
2194:
2191:
2188:
2185:
2182:
2178:
2174:
2037:
2031:
2017:
2016:
2005:
1998:
1994:
1988:
1982:
1976:
1969:
1965:
1959:
1953:
1947:
1944:
1917:
1914:
1811:
1810:
1799:
1796:
1791:
1788:
1783:
1780:
1777:
1774:
1769:
1766:
1761:
1758:
1755:
1752:
1747:
1743:
1739:
1736:
1679:, then either
1640:
1639:
1628:
1625:
1622:
1619:
1616:
1613:
1610:
1607:
1604:
1601:
1598:
1595:
1592:
1589:
1586:
1583:
1580:
1577:
1574:
1571:
1568:
1565:
1562:
1559:
1556:
1553:
1521:, the numbers
1449:
1448:
1437:
1434:
1431:
1428:
1425:
1422:
1419:
1416:
1413:
1410:
1407:
1404:
1401:
1398:
1395:
1392:
1336:is an element
1306:
1303:
1301:
1298:
1294:Iwasawa theory
1278:Richard Taylor
1215:
1212:
1180:
1177:
1165:Hilbert symbol
1128:
1125:
1079:
1076:
1008:
1005:
977:Disquisitiones
969:Disquisitiones
934:
931:
906:
903:
878: 1800 BC
854:
853:
839:
834:
830:
826:
821:
817:
813:
810:
800:
786:
783:
780:
777:
774:
735:
732:
730:
727:
650:
649:
647:
646:
639:
632:
624:
621:
620:
612:
611:
583:
582:
576:
570:
564:
552:
547:
546:
543:
542:
539:
538:
527:
522:
518:
514:
510:
491:
478:
473:
454:
441:
436:
424:-adic integers
417:
411:
402:
388:
387:
386:
385:
379:
373:
372:
371:
359:
353:
347:
341:
335:
317:
312:
311:
308:
307:
304:
303:
302:
301:
289:
288:
287:
281:
269:
268:
267:
249:
248:
247:
246:
234:
230:
226:
221:
217:
214:
200:
188:
167:
161:
155:
149:
135:
134:
128:
122:
108:
107:
101:
95:
89:
88:
87:
81:
69:
63:
51:
49:Basic concepts
48:
47:
44:
43:
26:
9:
6:
4:
3:
2:
7212:
7201:
7200:Number theory
7198:
7196:
7193:
7192:
7190:
7175:
7171:
7167:
7165:
7161:
7157:
7155:
7147:
7145:
7137:
7135:
7127:
7126:
7123:
7117:
7114:
7112:
7109:
7107:
7104:
7102:
7099:
7097:
7094:
7092:
7091:Modular forms
7089:
7087:
7084:
7083:
7081:
7077:
7071:
7068:
7066:
7063:
7061:
7058:
7056:
7053:
7050:
7046:
7043:
7041:
7038:
7036:
7033:
7031:
7028:
7026:
7023:
7021:
7018:
7016:
7015:Prime numbers
7013:
7011:
7008:
7006:
7003:
7001:
6998:
6996:
6993:
6992:
6990:
6986:
6980:
6977:
6975:
6972:
6969:
6965:
6961:
6958:
6955:
6951:
6948:
6945:
6941:
6937:
6934:
6932:
6929:
6927:
6924:
6922:
6919:
6916:
6912:
6908:
6904:
6901:
6898:
6897:Kummer theory
6894:
6890:
6886:
6882:
6878:
6875:
6874:
6872:
6868:
6864:
6863:Number theory
6857:
6852:
6850:
6845:
6843:
6838:
6837:
6834:
6826:
6822:
6821:
6816:
6812:
6809:
6804:
6800:
6799:
6789:
6785:
6781:
6777:
6773:
6767:
6763:
6758:
6753:
6749:
6745:
6742:
6738:
6734:
6728:
6724:
6720:
6716:
6712:
6708:
6705:
6701:
6697:
6695:0-521-43834-9
6691:
6687:
6683:
6679:
6675:
6671:
6668:
6664:
6660:
6656:
6652:
6648:
6647:
6638:
6632:
6628:
6627:
6626:Number Fields
6621:
6620:
6611:
6605:
6602:, CRC Press,
6601:
6600:
6595:
6591:
6587:
6584:
6578:
6574:
6570:
6566:
6561:
6555:
6554:
6548:
6547:
6533:
6529:
6525:
6521:
6517:
6511:
6507:
6503:
6499:
6495:
6494:
6483:
6476:
6470:
6464:
6450:
6446:
6442:
6436:
6429:
6425:
6421:
6417:
6411:
6396:
6392:
6385:
6378:
6376:1-85702-521-0
6372:
6368:
6367:
6362:
6356:
6349:
6345:
6341:
6337:
6333:
6329:
6328:Hasse, Helmut
6323:
6314:
6307:
6305:0-387-94674-8
6301:
6297:
6293:
6286:
6279:
6273:
6269:
6262:
6249:on 2021-05-22
6245:
6241:
6234:
6227:
6225:
6217:
6211:
6207:
6206:
6198:
6189:
6180:
6171:
6167:
6157:
6154:
6152:
6149:
6147:
6146:Kummer theory
6144:
6142:
6139:
6138:
6132:
6130:
6126:
6122:
6118:
6111:Related areas
6108:
6106:
6102:
6096:
6086:
6083:
6079:
6075:
6071:
6067:
6063:
6059:
6055:
6051:
6047:
6043:
6039:
6035:
6031:
6027:
6023:
6019:
6014:
6012:
6008:
5989:
5982:
5978:
5975:
5972:
5964:
5960:
5957:
5954:
5943:
5940:
5934:
5930:
5925:
5922:
5917:
5912:
5907:
5904:
5899:
5891:
5890:
5889:
5887:
5881:
5871:
5869:
5862:
5859: +
5855:
5852:
5848:
5844:
5840:
5833:
5826:
5819:
5816: +
5812:
5808:
5804:
5800:
5796:
5792:
5788:
5784:
5780:
5774:
5764:
5762:
5758:
5754:
5750:
5746:
5734:Major results
5731:
5728:
5712:
5707:
5696:
5690:
5686:
5677:
5673:
5669:
5665:
5661:
5657:
5653:
5648:
5638:
5636:
5631:
5629:
5624:
5619:
5615:
5611:
5606:
5601:
5597:
5592:
5586:
5580:
5575:
5571:
5564:Zeta function
5561:
5559:
5555:
5539:
5536:
5529:
5525:
5521:
5516:
5512:
5507:
5503:
5500:
5497:
5492:
5488:
5478:
5472:
5466:
5460:
5454:
5448:
5442:
5413:
5403:
5393:
5385:
5382:
5376:
5370:
5364:
5353:
5349:
5345:
5340:
5336:
5320:
5316:
5312:
5309:
5303:
5294:
5293:
5292:
5289:
5283:
5281:
5277:
5273:
5272:Galois module
5269:
5265:
5260:
5255:
5245:
5238:
5233:
5228:
5223:
5219:
5214:
5208:
5202:
5192:
5186:
5182:
5177:
5156:
5141:
5133:
5129:
5125:
5117:
5113:
5102:
5099:
5096:
5093:
5089:
5079:
5059:
5055:
5051:
5042:
5017:
5014:
5004:
4985:
4977:
4974:
4971:
4963:
4959:
4948:
4945:
4942:
4939:
4935:
4931:
4929:
4921:
4918:
4913:
4905:
4902:
4899:
4891:
4887:
4883:
4877:
4874:
4871:
4863:
4859:
4855:
4850:
4846:
4842:
4837:
4834:
4826:
4823:
4820:
4812:
4809:
4805:
4801:
4798:
4795:
4790:
4787:
4779:
4776:
4773:
4765:
4762:
4758:
4743:
4729:
4726:
4723:
4715:
4693:
4690:
4687:
4678:
4656:
4652:
4631:
4608:
4602:
4579:
4573:
4565:
4562:measures the
4547:
4543:
4520:
4510:
4507:
4487:
4484:
4479:
4475:
4451:
4448:
4445:
4439:
4436:
4416:
4409:at the point
4393:
4387:
4383:
4376:
4370:
4350:
4347:
4344:
4322:
4318:
4297:
4277:
4257:
4234:
4228:
4206:
4196:
4193:
4184:
4170:
4150:
4124:
4118:
4115:
4093:
4083:
4074:
4051:
4028:
4022:
4019:
4016:
4009:
4005:
4001:
3997:
3981:
3977:
3973:
3951:
3941:
3938:
3924:
3922:
3917:
3912:
3895:
3892:
3883:
3879:to mean that
3863:
3860:
3851:
3846:
3842:
3841:finite primes
3838:
3830:
3821:
3816:
3812:
3808:
3804:
3800:
3799:finite places
3796:
3792:
3740:
3737:functions on
3736:
3732:
3727:
3725:
3721:
3717:
3713:
3709:
3705:
3701:
3694:functions |·|
3693:
3689:
3685:
3681:
3677:
3667:
3630:
3624:
3614:
3609:
3604:
3601:
3594:
3589:
3586:-dimensional
3584:
3578:
3574:The image of
3572:
3555:
3552:
3546:
3543:
3540:
3534:
3531:
3528:
3517:
3513:
3508:
3504:
3498:
3493:
3488:
3482:
3480:
3464:
3457:
3453:
3449:
3439:
3432:
3428:
3414:
3411:
3408:
3384:
3380:
3369:
3362:
3358:
3344:
3341:
3338:
3329:
3326:
3320:
3314:
3307:
3300:
3290:
3283:
3277:
3273:
3265:
3256:
3250:
3244:
3240:
3232:
3226:
3216:
3205:
3199:
3192:
3183:
3174:
3167:
3162:
3155:
3151:
3147:
3139:
3134:
3128:
3126:
3122:
3117:
3113:
3107:
3103:
3092:
3081:
3056:
3050:
3047:
3044:
3038:
3035:
3032:
3022:
3015:
3011:
3002:
2998:
2991:
2984:
2983:
2982:
2980:
2976:
2972:
2967:
2958:
2939:
2931:
2914:
2910:
2904:
2899:
2896:
2893:
2889:
2885:
2882:
2879:
2876:
2869:
2868:
2867:
2864:
2843:
2836:
2832:
2826:
2814:
2807:
2803:
2797:
2785:
2779:
2769:
2768:
2767:
2764:
2760:
2754:
2747:
2741:
2736:
2733:. There is a
2731:
2726:
2721:
2715:
2710:
2694:
2689:
2685:
2680:
2678:
2673:
2666:
2659:
2652:
2648:
2642:
2638:
2632:
2626:
2620:
2616:
2610:
2590:
2584:
2581:
2578:
2575:
2572:
2569:
2566:
2563:
2557:
2551:
2548:
2545:
2539:
2534:
2531:
2527:
2519:
2518:
2517:
2515:
2510:
2504:
2497:
2491:
2485:
2479:
2473:
2467:
2461:
2457:
2451:
2447:
2441:
2435:
2429:
2424:
2420:
2409:
2407:
2403:
2399:
2395:
2391:
2385:
2381:
2373:
2366:
2360:
2357:
2351:
2346:
2341:
2334:
2327:
2323:
2320:= (1 −
2319:
2298:
2293:
2282:
2268:
2265:
2262:
2253:
2247:
2233:
2230:
2227:
2221:
2215:
2201:
2198:
2195:
2189:
2183:
2172:
2165:
2164:
2163:
2162:implies that
2159:
2155:
2148:
2145:
2139:
2133:
2130:
2124:
2122:
2117:
2111:
2106:
2101:
2095:
2089:
2078:
2072:
2066:
2064:
2058:
2053:
2035:
2003:
1996:
1992:
1986:
1974:
1967:
1963:
1957:
1945:
1942:
1935:
1934:
1933:
1930:
1924:
1913:
1910:
1866:. Similarly,
1860:
1856:
1797:
1789:
1786:
1781:
1778:
1767:
1764:
1759:
1756:
1750:
1745:
1741:
1737:
1734:
1727:
1726:
1725:
1700:
1694:
1689:
1683:
1677:
1673:
1667:
1663:
1659:
1653:
1651:
1646:
1626:
1620:
1617:
1614:
1605:
1602:
1599:
1593:
1587:
1584:
1581:
1578:
1569:
1566:
1563:
1560:
1554:
1551:
1544:
1543:
1542:
1539:
1533:
1526:
1519:
1515:
1511:
1506:
1499:
1494:
1489:
1483:
1477:
1471:
1465:
1460:
1455:
1435:
1429:
1426:
1420:
1414:
1411:
1405:
1402:
1399:
1396:
1393:
1390:
1383:
1382:
1381:
1370:
1364:
1358:
1352:
1348:such that if
1346:
1340:
1335:
1334:prime element
1330:
1327:
1321:
1316:
1315:prime numbers
1312:
1300:Basic notions
1297:
1295:
1291:
1287:
1283:
1279:
1275:
1271:
1267:
1263:
1258:
1256:
1252:
1247:
1245:
1241:
1237:
1233:
1232:modular forms
1229:
1225:
1221:
1214:Modern theory
1211:
1209:
1205:
1201:
1197:
1193:
1189:
1185:
1176:
1174:
1170:
1166:
1162:
1158:
1153:
1151:
1147:
1143:
1139:
1138:
1133:
1132:David Hilbert
1124:
1122:
1121:ideal numbers
1118:
1114:
1110:
1106:
1102:
1096:
1092:
1090:
1089:
1084:
1075:
1073:
1069:
1065:
1061:
1057:
1053:
1049:
1044:
1042:
1038:
1034:
1030:
1029:number fields
1026:
1022:
1018:
1014:
1004:
1002:
998:
994:
990:
986:
982:
978:
973:
970:
966:
962:
958:
954:
950:
946:
942:
941:
930:
928:
923:
919:
915:
911:
902:
900:
899:
893:
891:
887:
883:
872:
868:
864:
860:
837:
832:
828:
824:
819:
815:
811:
808:
801:
784:
781:
778:
775:
772:
765:
764:
763:
761:
757:
753:
749:
745:
741:
726:
724:
720:
716:
715:Galois groups
712:
708:
704:
700:
696:
695:finite fields
692:
688:
684:
680:
677:to study the
676:
672:
671:number theory
668:
661:
656:
645:
640:
638:
633:
631:
626:
625:
623:
622:
617:
616:
610:
606:
605:
604:
603:
602:
597:
596:
595:
590:
589:
588:
581:
577:
575:
571:
569:
565:
563:
562:Division ring
559:
558:
557:
556:
550:
545:
544:
516:
500:
498:
492:
476:
462:
461:-adic numbers
460:
455:
439:
425:
423:
418:
416:
412:
410:
403:
401:
397:
396:
395:
394:
393:
384:
380:
378:
374:
370:
366:
365:
364:
360:
358:
354:
352:
348:
346:
342:
340:
336:
334:
330:
329:
328:
324:
323:
322:
321:
315:
310:
309:
300:
296:
295:
294:
290:
286:
282:
280:
276:
275:
274:
270:
266:
262:
261:
260:
256:
255:
254:
253:
228:
224:
215:
212:
205:
204:Terminal ring
201:
178:
174:
173:
172:
168:
166:
162:
160:
156:
154:
150:
148:
144:
143:
142:
141:
140:
133:
129:
127:
123:
121:
117:
116:
115:
114:
113:
106:
102:
100:
96:
94:
90:
86:
82:
80:
76:
75:
74:
73:Quotient ring
70:
68:
64:
62:
58:
57:
56:
55:
46:
45:
42:
37:→ Ring theory
36:
32:
31:
19:
6988:Key concepts
6915:sieve theory
6876:
6818:
6751:
6714:
6681:
6658:
6625:
6598:
6590:Stewart, Ian
6564:
6552:
6505:
6501:
6475:
6463:
6451:the element
6448:
6440:
6435:
6427:
6426:the element
6423:
6415:
6410:
6398:. Retrieved
6394:
6384:
6364:
6361:Singh, Simon
6355:
6339:
6322:
6313:
6291:
6285:
6267:
6261:
6251:, retrieved
6244:the original
6239:
6208:, Springer,
6204:
6197:
6188:
6179:
6170:
6120:
6114:
6105:number field
6100:
6098:
6073:
6069:
6065:
6061:
6057:
6053:
6049:
6045:
6037:
6034:prime number
6025:
6021:
6015:
6006:
6004:
5883:
5867:
5860:
5853:
5842:
5838:
5831:
5824:
5817:
5810:
5806:
5802:
5798:
5794:
5790:
5786:
5782:
5778:
5776:
5760:
5757:class number
5744:
5742:
5675:
5671:
5667:
5659:
5655:
5650:
5641:Local fields
5632:
5622:
5604:
5590:
5584:
5578:
5567:
5553:
5476:
5470:
5464:
5458:
5452:
5446:
5440:
5437:
5287:
5284:
5279:
5275:
5267:
5263:
5258:
5253:
5243:
5236:
5226:
5224:that lie in
5212:
5206:
5203:
5190:
5184:
5175:
5163:
5081:
5006:
4745:
4714:power series
4185:
3930:
3913:
3881:
3849:
3844:
3840:
3836:
3828:
3819:
3814:
3813:. These are
3810:
3806:
3802:
3798:
3794:
3790:
3738:
3730:
3728:
3723:
3719:
3711:
3707:
3703:
3699:
3690:, there are
3687:
3683:
3673:
3628:
3622:
3612:
3608:discriminant
3607:
3602:
3599:
3592:
3582:
3576:
3573:
3506:
3502:
3496:
3486:
3483:
3478:
3330:
3324:
3318:
3312:
3305:
3298:
3288:
3281:
3278:is the pair
3275:
3271:
3263:
3254:
3248:
3245:
3238:
3224:
3214:
3203:
3197:
3190:
3181:
3172:
3160:
3153:
3149:
3145:
3137:
3132:
3129:
3124:
3120:
3115:
3111:
3105:
3101:
3090:
3079:
3076:
2969:, while the
2965:
2954:
2862:
2858:
2762:
2758:
2752:
2745:
2739:
2729:
2719:
2711:
2692:
2687:
2684:class number
2683:
2681:
2677:Picard group
2671:
2664:
2657:
2650:
2646:
2640:
2636:
2630:
2624:
2618:
2614:
2608:
2605:
2508:
2502:
2495:
2489:
2483:
2477:
2471:
2465:
2459:
2455:
2449:
2445:
2439:
2433:
2427:
2415:
2400:(that is, a
2397:
2389:
2383:
2379:
2371:
2364:
2358:
2355:
2349:
2339:
2332:
2325:
2321:
2317:
2313:
2157:
2156:)(1 −
2153:
2146:
2143:
2137:
2131:
2128:
2125:
2115:
2109:
2099:
2093:
2090:
2076:
2070:
2067:
2056:
2018:
1928:
1922:
1919:
1908:
1858:
1854:
1812:
1698:
1692:
1687:
1681:
1675:
1671:
1665:
1661:
1654:
1644:
1641:
1537:
1531:
1524:
1517:
1509:
1504:
1497:
1492:
1487:
1481:
1475:
1469:
1463:
1453:
1450:
1368:
1362:
1356:
1350:
1344:
1338:
1333:
1331:
1325:
1319:
1308:
1262:Andrew Wiles
1259:
1248:
1244:modular form
1220:Goro Shimura
1217:
1182:
1173:Teiji Takagi
1154:
1135:
1130:
1117:Emmy Noether
1100:
1098:
1094:
1086:
1081:
1063:
1059:
1045:
1010:
985:Ernst Kummer
976:
974:
968:
948:
939:
938:
936:
921:
908:
896:
894:
885:
881:
866:
862:
858:
855:
759:
755:
751:
747:
737:
666:
665:
613:
599:
598:
594:Free algebra
592:
591:
585:
584:
553:
496:
458:
421:
391:
390:
389:
369:Finite field
318:
265:Finite field
251:
250:
177:Initial ring
137:
136:
110:
109:
52:
7174:Wikiversity
7096:L-functions
6711:Lang, Serge
6594:Tall, David
5658:at a place
5647:Local field
4337:at a point
3516:dot product
3252:is denoted
2374:≡ 1 (mod 4)
2367:≡ 3 (mod 4)
2052:prime ideal
2019:where each
1530:−2 +
1204:norm symbol
1163:and of the
1137:Zahlbericht
1105:ring theory
1101:Vorlesungen
997:L-functions
922:Arithmetica
914:conjectured
898:Arithmetica
740:Alexandrian
574:Simple ring
285:Jordan ring
159:Graded ring
41:Ring theory
7189:Categories
7055:Arithmetic
6788:0956.11021
6532:0948.11001
6400:21 January
6253:2007-12-25
5652:Completing
5210:, denoted
4000:projective
3916:adele ring
3676:valuations
2766:satisfies
2644:such that
2338:1 −
1251:André Weil
1200:Eisenstein
1184:Emil Artin
1033:unit group
912:was first
744:Diophantus
734:Diophantus
713:, and the
689:and their
580:Commutator
339:GCD domain
6825:EMS Press
6445:adjoining
6420:adjoining
5976:−
5958:−
5941:−
5554:regulator
5501:⋯
5386:
5326:→
5321:×
5249:− 1
5108:∞
5100:−
5090:∑
5018:∈
5007:for some
4975:−
4954:∞
4946:−
4936:∑
4922:⋯
4903:−
4875:−
4835:−
4824:−
4810:−
4799:⋯
4788:−
4777:−
4763:−
4727:−
4691:−
4511:∈
4485:≠
4348:∈
4128:^
4119:−
4084:⊂
4078:^
3899:∞
3896:∤
3867:∞
3864:∣
3647:Δ
3547:
3538:⟩
3526:⟨
3440:⊕
3418:→
3412::
3370:⊕
3348:→
3342::
3272:signature
3054:→
3048:
3042:→
3036:
3016:×
3008:→
3003:×
2995:→
2890:∑
2880:
2815:⋯
2703:(2, 1 + √
2582:⊆
2567:∈
2532:−
2231:−
2222:⋅
2152:2 = (1 +
2082:(2, 1 + √
1975:⋯
1787:−
1782:−
1765:−
1618:−
1582:−
1493:associate
1467:, and if
1427:−
1421:⋅
1412:−
1400:⋅
1007:Dirichlet
521:∞
299:Semifield
7164:Wikibook
7134:Category
6750:(1999).
6713:(1994),
6680:(1993),
6596:(2015),
6363:(1997),
6338:(eds.),
6296:Springer
6135:See also
5797:, where
5662:gives a
5170:−1
4671:is then
4183:-adics.
3698: :
3316:, where
3235:−√
3188:−√
3095:−1
3023:→
2971:cokernel
2714:divisors
1378:−2
1286:category
1078:Dedekind
981:European
965:Legendre
961:Lagrange
679:integers
293:Semiring
279:Lie ring
61:Subrings
6995:Numbers
6827:, 2001
6780:1697859
6741:1282723
6704:1215934
6667:0215665
6524:1737196
6480:Stein.
6348:0215665
6292:Hilbert
6052:) and (
5616:of the
5582:. When
5450:and |·|
3606:is the
3588:lattice
3490:called
3237:−
3223:−
3213:−
3202:−
3186:and to
3143:, with
2406:abelian
1823:)(2 - √
1503:−
1290:schemes
1240:modular
1127:Hilbert
1109:Hilbert
705:admits
495:Prüfer
97:•
6870:Fields
6786:
6778:
6768:
6739:
6729:
6702:
6692:
6665:
6633:
6606:
6579:
6530:
6522:
6512:
6373:
6346:
6302:
6274:
6212:
6042:modulo
6036:is an
5805:, and
5438:where
5179:. The
4250:where
4006:. The
3996:smooth
3837:primes
3670:Places
3618:Δ
3270:. The
3164:not a
3158:, and
3156:> 0
2957:kernel
2612:where
2501:(1) =
2392:is an
1819:(2 + √
1070:. The
1019:, for
905:Fermat
841:
788:
719:fields
711:ideals
697:, and
147:Module
120:Kernel
7010:Unity
6557:(PDF)
6485:(PDF)
6247:(PDF)
6236:(PDF)
6162:Notes
6078:Artin
5845:is a
5608:is a
5270:as a
5196:2 + √
5160:Units
3994:be a
3847:. If
3731:place
3590:. If
2859:Then
2737:from
2668:, or
2419:group
2404:with
2378:(1 +
2068:When
2050:is a
1901:2 - √
1894:2 + √
1875:2 - √
1868:2 + √
1842:2 - √
1835:2 + √
1829:. If
1827:) = 9
1715:2 - √
1708:2 + √
1660:. An
1523:1 + 2
1457:is a
1179:Artin
957:Euler
945:Latin
933:Gauss
499:-ring
363:Field
259:Field
67:Ideal
54:Rings
6766:ISBN
6727:ISBN
6690:ISBN
6631:ISBN
6604:ISBN
6577:ISBN
6510:ISBN
6402:2013
6371:ISBN
6300:ISBN
6272:ISBN
6210:ISBN
6099:The
5851:rank
5568:The
5168:and
3966:and
3598:det
3123:and
2955:The
2861:div
2751:Div
2718:Div
2670:Pic
2628:and
2475:and
2336:and
2331:1 +
2324:) ⋅
2316:1 +
1899:and
1873:and
1713:and
1528:and
1501:and
1485:and
1459:unit
1292:and
1268:for
1230:and
1222:and
999:and
991:and
975:The
963:and
884:+ 65
758:and
750:and
703:ring
6784:Zbl
6569:doi
6528:Zbl
6447:to
6443:by
6422:to
6418:by
5849:of
5670:or
5383:log
4624:at
4566:of
3809:or
3789:of
3632:is
3620:or
3610:of
3304:+ 2
3274:of
3219:to
3177:to
3109:or
3033:Div
3027:div
2961:div
2959:of
2877:div
2686:of
2663:Cl
2656:Cl
2453:if
2431:of
2396:of
2097:of
1920:If
1857:+ 3
1840:or
1685:or
1366:or
1342:of
1288:of
1167:of
1035:of
916:by
717:of
7191::
6966:,
6942:,
6913:,
6909:,
6895:,
6891:,
6887:,
6883:,
6823:,
6817:,
6782:.
6776:MR
6774:.
6764:.
6754:.
6737:MR
6735:,
6725:,
6717:,
6700:MR
6698:,
6688:,
6676:;
6663:MR
6653:;
6592:;
6575:,
6526:,
6520:MR
6518:,
6504:,
6455:-5
6393:.
6344:MR
6334:;
6298:,
6294:,
6238:,
6223:^
6131:.
6013:.
6005:A
5870:.
5809:=
5793:×
5763:.
5637:.
5540:0.
5242:+
4002:,
3998:,
3923:.
3729:A
3714:.
3702:→
3686:→
3666:.
3571:.
3544:Tr
3505:∈
3481:.
3328:.
3311:=
3287:,
3243:.
3233:,
3200:(√
3152:,
3148:∈
3135:(√
3114:→
3104:→
3093:(√
3082:(√
3057:1.
3045:Cl
2761:∈
2709:.
2705:-5
2697:-5
2695:(√
2661:,
2649:=
2647:xI
2639:∈
2617:∈
2609:Ox
2516::
2484:IJ
2458:∈
2448:⊆
2446:xJ
2353:,
2084:-5
2065:.
1903:-5
1896:-5
1892:,
1877:-5
1870:-5
1863:-5
1844:-5
1837:-5
1825:-5
1821:-5
1717:-5
1710:-5
1706:,
1676:yz
1674:=
1488:up
1476:up
1357:ab
1332:A
1329:.
1246:.
1210:.
1175:.
987:,
959:,
947::
875:c.
865:=
861:+
725:.
693:,
681:,
607:•
578:•
572:•
566:•
560:•
493:•
456:•
419:•
413:•
404:•
398:•
381:•
375:•
367:•
361:•
355:•
349:•
343:•
337:•
331:•
325:•
297:•
291:•
283:•
277:•
271:•
263:•
257:•
202:•
175:•
169:•
163:•
157:•
151:•
145:•
130:•
124:•
118:•
103:•
91:•
83:•
77:•
71:•
65:•
59:•
7051:)
7047:(
7000:0
6970:)
6962:(
6956:)
6952:(
6946:)
6938:(
6917:)
6905:(
6899:)
6879:(
6855:e
6848:t
6841:v
6790:.
6571::
6487:.
6458:.
6453:√
6449:Z
6441:Z
6430:.
6428:i
6424:Z
6416:Z
6404:.
6121:Z
6074:p
6072:/
6070:b
6068:,
6066:a
6062:p
6058:p
6056:/
6054:q
6050:q
6048:/
6046:p
6038:n
6026:q
6024:/
6022:p
6020:(
5990:.
5983:2
5979:1
5973:q
5965:2
5961:1
5955:p
5948:)
5944:1
5938:(
5935:=
5931:)
5926:p
5923:q
5918:(
5913:)
5908:q
5905:p
5900:(
5868:O
5864:2
5861:r
5857:1
5854:r
5843:O
5839:K
5835:2
5832:r
5828:1
5825:r
5821:2
5818:r
5814:1
5811:r
5807:r
5803:O
5799:G
5795:Z
5791:G
5787:O
5783:O
5779:O
5761:h
5745:K
5713::
5708:p
5703:Q
5697:/
5691:w
5687:K
5676:p
5672:C
5668:R
5660:w
5656:K
5623:K
5605:K
5591:Q
5585:K
5579:K
5537:=
5530:2
5526:r
5522:+
5517:1
5513:r
5508:x
5504:+
5498:+
5493:1
5489:x
5477:O
5471:K
5465:L
5459:v
5453:v
5447:K
5441:v
5414:v
5410:)
5404:v
5399:|
5394:x
5390:|
5380:(
5377:=
5374:)
5371:x
5368:(
5365:L
5354:2
5350:r
5346:+
5341:1
5337:r
5331:R
5317:K
5313::
5310:L
5304:{
5288:K
5280:Q
5278:/
5276:K
5268:Q
5264:Z
5261:⊗
5259:O
5254:Q
5247:2
5244:r
5240:1
5237:r
5227:O
5213:O
5207:O
5198:3
5191:Z
5185:Z
5176:i
5174:±
5166:1
5142:n
5138:)
5134:t
5130:/
5126:1
5123:(
5118:n
5114:a
5103:k
5097:=
5094:n
5066:)
5063:)
5060:t
5056:/
5052:1
5049:(
5046:(
5043:k
5022:N
5015:k
4986:n
4982:)
4978:2
4972:t
4969:(
4964:n
4960:a
4949:k
4943:=
4940:n
4932:=
4919:+
4914:2
4910:)
4906:2
4900:t
4897:(
4892:2
4888:a
4884:+
4881:)
4878:2
4872:t
4869:(
4864:1
4860:a
4856:+
4851:0
4847:a
4843:+
4838:1
4831:)
4827:2
4821:t
4818:(
4813:1
4806:a
4802:+
4796:+
4791:k
4784:)
4780:2
4774:t
4771:(
4766:k
4759:a
4730:2
4724:t
4700:)
4697:)
4694:2
4688:t
4685:(
4682:(
4679:k
4657:2
4653:v
4632:2
4612:)
4609:x
4606:(
4603:q
4583:)
4580:x
4577:(
4574:p
4548:2
4544:v
4521:1
4516:A
4508:2
4488:0
4480:1
4476:x
4455:]
4452:1
4449::
4446:2
4443:[
4440:=
4437:p
4417:p
4397:)
4394:x
4391:(
4388:q
4384:/
4380:)
4377:x
4374:(
4371:p
4351:X
4345:p
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4319:v
4298:t
4278:F
4258:t
4238:)
4235:t
4232:(
4229:k
4207:1
4202:P
4197:=
4194:X
4171:p
4151:F
4125:X
4116:X
4094:n
4089:A
4075:X
4052:X
4032:)
4029:X
4026:(
4023:k
4020:=
4017:F
3982:k
3978:/
3974:X
3952:q
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3942:=
3939:k
3893:v
3882:v
3861:v
3850:v
3832:2
3829:r
3823:1
3820:r
3811:C
3807:R
3803:K
3795:p
3791:O
3775:p
3751:p
3739:K
3724:Q
3720:Q
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3708:p
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3700:Q
3696:p
3688:R
3684:Q
3651:|
3643:|
3629:O
3623:D
3613:O
3603:B
3600:B
3593:B
3583:d
3577:O
3559:)
3556:y
3553:x
3550:(
3541:=
3535:y
3532:,
3529:x
3507:K
3503:x
3497:K
3487:d
3465:.
3458:2
3454:r
3450:2
3445:R
3433:1
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3423:R
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3409:M
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3292:2
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3285:1
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3280:(
3276:K
3267:2
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3255:r
3249:K
3239:a
3225:a
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3141:)
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2940:.
2937:]
2932:i
2926:p
2920:[
2915:i
2911:e
2905:t
2900:1
2897:=
2894:i
2886:=
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2863:x
2844:.
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2808:1
2804:e
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2792:p
2786:=
2783:)
2780:x
2777:(
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2588:}
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2528:J
2509:J
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2326:i
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2318:i
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2290:)
2286:]
2283:i
2280:[
2276:Z
2272:)
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2260:(
2257:(
2254:=
2251:]
2248:i
2245:[
2241:Z
2237:)
2234:i
2228:1
2225:(
2219:]
2216:i
2213:[
2209:Z
2205:)
2202:i
2199:+
2196:1
2193:(
2190:=
2187:]
2184:i
2181:[
2177:Z
2173:2
2160:)
2158:i
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2138:Z
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2116:E
2110:O
2100:K
2094:E
2086:)
2077:Z
2071:O
2057:I
2036:i
2030:p
2004:,
1997:t
1993:e
1987:t
1981:p
1968:1
1964:e
1958:1
1952:p
1946:=
1943:I
1929:O
1923:I
1909:Z
1890:3
1886:3
1882:3
1861:√
1859:b
1855:a
1853:3
1849:3
1831:3
1815:3
1798:.
1795:)
1790:5
1779:2
1776:(
1773:)
1768:5
1760:+
1757:2
1754:(
1751:=
1746:2
1742:3
1738:=
1735:9
1722:9
1704:3
1699:Z
1693:O
1688:z
1682:y
1672:x
1666:x
1645:Z
1627:,
1624:)
1621:i
1615:2
1612:(
1609:)
1606:i
1603:+
1600:2
1597:(
1594:=
1591:)
1588:i
1585:2
1579:1
1576:(
1573:)
1570:i
1567:2
1564:+
1561:1
1558:(
1555:=
1552:5
1538:i
1532:i
1525:i
1518:Z
1510:K
1505:p
1498:p
1482:p
1470:p
1464:O
1454:u
1436:.
1433:)
1430:3
1424:(
1418:)
1415:2
1409:(
1406:=
1403:3
1397:2
1394:=
1391:6
1374:1
1369:b
1363:a
1351:p
1345:O
1339:p
1326:K
1320:O
1064:n
1060:n
943:(
886:y
882:x
867:z
863:y
859:x
838:.
833:2
829:y
825:+
820:2
816:x
812:=
809:B
785:y
782:+
779:x
776:=
773:A
760:B
756:A
752:y
748:x
643:e
636:t
629:v
526:)
517:p
513:(
509:Z
497:p
477:p
472:Q
459:p
440:p
435:Z
422:p
408:n
233:Z
229:1
225:/
220:Z
216:=
213:0
187:Z
20:)
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