220:“must be granted a fully simultaneous treatment instead of the segregated status, and at best the separate but equal facilities, which hitherto have been their lot. That, far from losing by such treatment, both races stand to gain by it, is one fact which will, I hope, clearly emerge from this book.”
377:
for infinitely many places of K. This approach also allows for a significantly simpler and more logical proof of algebraic statements, for example the result that a simple algebra over an A-field splits (globally) if and only if it splits everywhere locally. The systematic use of simple algebras also
364:
in particular. The author acknowledges this as a trade-off, explaining that “to develop such an approach systematically would have meant loading a great deal of unnecessary machinery on a ship which seemed well equipped for this particular voyage; instead of making it more seaworthy, it might have
306:
in order to describe global class field theory for infinite extensions, but several years later he used it in a new way to derive global class field theory from local class field theory. Weil mentioned this (unpublished) work as a significant influence on some of the choices of treatment he uses.
335:
showed me the first edition in autumn 1967 in Moscow and said that this book will be from now on the book about class field theory". The coherence of the treatment, and some of its distinctive features, were highlighted by several reviewers, with Koch going on to say "This book is written in the
215:
ring in which all the valuations are brought together in a single coherent way in which they “cooperate for a common purpose”. Removing the real numbers from a pedestal and placing them alongside the p-adic numbers leads naturally – “it goes without saying” to the development of the theory of
413:
are used to prove that `local fields’ (commutative fields locally compact under a non-discrete topology) are completions of A-fields. In particular – a concept developed later – they are precisely the fields whose local class field theory is needed for the global theory. The non-discrete
216:
function fields over finite fields in a “fully simultaneous treatment with number-fields”. In a striking choice of wording for a foreword written in the United States in 1967, the author chooses to drive this particular viewpoint home by explaining that the two classes of
134:
over local and global fields. The word `basic’ in the title is closer in meaning to `foundational’ rather than `elementary’, and is perhaps best interpreted as meaning that the material developed is foundational for the development of the theories of
365:
sunk it.” The treatment of class field theory uses analytic methods on both commutative fields and simple algebras. These methods show their power in giving the first unified proof that if K/k is a finite
1591:
Algebraic number theory : proceedings of an instructional conference organized by the London
Mathematical Society (a NATO advanced study institute) with the support of the International Mathematical
147:, and more advanced topics in algebraic number theory. The style is austere, with a narrow concentration on a logically coherent development of the theory required, and essentially no examples.
348:
lic methods and the simultaneous treatment of algebraic number fields and rational function fields over finite fields. The second half is arguably pre-modern in its development of
159:
classical treatment of algebraic number theory, he “rather tried to draw the conclusions from the developments of the last thirty years, whereby locally compact groups,
1381:
884:
336:
spirit of the early forties and just this makes it a valuable source of information for everyone who is working about problems related to number and function fields."
1385:
888:
163:
and integration have been seen to play an increasingly important role in classical number theory”. Weil goes on to explain a viewpoint that grew from work of
426:
Finite-dimensional vector spaces over local fields and division algebras under the topology uniquely determined by the field's topology are studied, and
520:
511:
This chapter departs slightly from the simultaneous treatment of number fields and function fields. In the number field setting, lattices (that is,
699:
Some references are added, some minor corrections made, some comments added, and five appendices are included, containing the following material:
683:
The global class field theory for A-fields is developed using the pairings of
Chapter XII, replacing multiplicative groups of local fields with
1599:
804:
763:
58:
324:
1594:. Cassels, J. W. S. (John William Scott), Fröhlich, A. (Albrecht), 1916- (2nd ed.). London: London Mathematical Society. 2010.
1357:
860:
993:
903:
92:
1686:
576:
248:
438:
of these vector spaces, which in the commutative one-dimensional case reduces to `self duality’ for local fields, are shown.
406:
688:
627:
The zeta-function of a simple algebra over an A-field is defined, and used to prove further results on the norm group and
949:
Hasse, Helmut (1930-01-01). "Die
Normenresttheorie relativ-Abelscher Zahlkörper als Klassenkörpertheorie im Kleinen".
496:
1681:
1623:
587:
to the whole plane, along with its functional equation. The treatment here goes back ultimately to a suggestion of
721:
275:
260:
184:
532:
648:
572:
379:
279:
236:
76:
838:
Vorlesungen ĂĽber die
Theorie der algebraischen Zahlen (Second edition of the 1923 original, with an index)
584:
431:
374:
267:
243:
formalism became a more significant part of local and global class field theory, particularly in work of
1572:
660:
580:
454:
212:
196:
160:
140:
123:
118:
of constants. The theory is developed in a uniform way, starting with topological fields, properties of
1232:
155:
In the foreword, the author explains that instead of the “futile and impossible task” of improving on
703:
A character version of the (local) transfer theorem and its extension to the global transfer theorem.
652:
316:
88:
382:. For instance, it is more straightforward to prove that a simple algebra over a local field has an
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103:
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1021:
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353:
244:
228:
80:
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A brief treatment of simple algebras is given, including explicit rules for cyclic factor sets.
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1595:
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856:
800:
759:
672:
664:
568:
536:
491:, with similar properties for finite-dimensional vector spaces and algebras over local fields.
484:
the A-field, when embedded diagonally, is a discrete and co-compact subring of its adele ring;
402:
361:
328:
320:
111:
53:
1039:
Artin, Emil (1929-12-01). "Idealklassen in oberkörpern und allgemeines reziprozitätsgesetz".
344:
Roughly speaking, the first half of the book is modern in its consistent use of adelic and id
1518:
1476:
1468:
1412:
1320:
1281:
1244:
1203:
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792:
751:
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592:
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427:
415:
366:
357:
332:
303:
283:
271:
208:
180:
172:
136:
107:
1451:
563:(and similar analytic objects) for an A-field are expressed in terms of integrals over the
1480:
1472:
1448:
656:
644:
540:
435:
386:
239:) which are defined in terms of the number field in proofs of class field theory. Instead
188:
144:
1493:
687:
class groups of A-fields. The pairing is constructed as a product over places of local
446:
349:
232:
131:
1675:
1538:
1434:
1155:
Tate, John (1952). "The Higher
Dimensional Cohomology Groups of Class Field Theory".
1068:
1025:
978:
935:
632:
556:
252:
240:
200:
72:
28:
710:
410:
370:
299:
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224:
217:
168:
119:
115:
99:
1308:
1191:
1576:
487:
the adele ring is self dual, meaning that it is topologically isomorphic to its
295:
192:
164:
156:
567:
group. Decomposing these integrals into products over all valuations and using
449:
are used to study extensions of the places of an A-field to places of a finite
1417:
1401:"Travaux de Claude Chevalley sur la théorie du corps de classes: Introduction"
1400:
1009:
962:
919:
796:
755:
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588:
560:
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466:
398:
383:
256:
204:
176:
127:
1609:
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1426:
1334:
1258:
1249:
1217:
1060:
1017:
970:
927:
1509:
Witt, Ernst (1931-12-01). "Über die kommutativität endlicher schiefkörper".
1367:
1325:
1208:
904:"Führer, Diskriminante und Verzweigungskörper relativ-Abelscher Zahlkörper"
870:
1272:
Hochschild, G.; Nakayama, T. (1952). "Cohomology in Class Field Theory".
684:
564:
478:
470:
287:
1656:
Sen, Shankar; Tate, John (1963). "Ramification groups of local fields".
1192:"Sur la Théorie du Corps de Classes sur le Corps des Nombres Rationnels"
1522:
1293:
1176:
1138:
1103:
1052:
199:
of the rationals, with no logical reason to favour it over the various
820:
1352:. Providence, R.I.: AMS Chelsea Pub./American Mathematical Society.
1285:
1168:
1130:
1095:
389:
than to prove the corresponding statement for 2-cohomology classes.
1511:
1041:
130:
and idelic number theory, and class field theory via the theory of
543:
defined as the class of divisors of non-trivial characters of the
1117:
Iwasawa, Kenkichi (1959). "Sheaves for
Algebraic Number Fields".
473:
group of an A-field, and proves the `main theorems’ as follows:
1637:
Shafarevich, Igor (1946). "On Galois groups of y-adic fields".
1082:
Iwasawa, Kenkichi (1953). "On the Rings of
Valuation Vectors".
274:
is particularly emphasised. In order to derive the theorems of
611:
of an algebraic extension of a function field are developed.
331:; in his review of the second edition Koch makes the remark "
323:. Later editions were reviewed by Fernando Q. GouvĂŞa for the
1569:
Fourier
Analysis in Number Fields and Hecke's Zeta-Functions
1237:
Japanese Journal of Mathematics: Transactions and Abstracts
603:
Formulas for local and global different and discriminants,
286:
introduced what he called the élément idéal, later called
270:
alongside function fields over finite fields, the work of
531:
This chapter is focused on the function field case; the
87:-theoretic methods. Based in part on a course taught at
434:
is proved in this context, and the main theorems about
651:
over a local field in the context of a pairing of the
1571:(Doctor of Philosophy thesis). Princeton University.
1233:"On the Theory of Ramification Groups and Conductors"
195:
may be seen as but one of infinitely many different
405:
proof that a finite division ring is commutative ('
315:The 1st edition was reviewed by George Whaples for
52:
42:
34:
24:
791:. Berlin, Heidelberg: Springer Berlin Heidelberg.
750:. Berlin, Heidelberg: Springer Berlin Heidelberg.
519:for a lattice is found. This is used to study the
515:) are defined, and the Haar measure volume of a
414:non-commutative locally compact fields are then
998:Journal fĂĽr die reine und angewandte Mathematik
951:Journal fĂĽr die reine und angewandte Mathematik
908:Journal fĂĽr die reine und angewandte Mathematik
98:series. The approach handles all 'A-fields' or
1554:Geometrie der Zahlen. In 2 Lieferungen. Lfg. 1
822:Grundlehren der mathematischen Wissenschaften
96:Grundlehren der mathematischen Wissenschaften
8:
1380:: CS1 maint: multiple names: authors list (
1313:Journal of the Mathematical Society of Japan
1196:Journal of the Mathematical Society of Japan
883:: CS1 maint: multiple names: authors list (
19:
994:"La théorie du symbole de restes normiques"
853:Lectures on the theory of algebraic numbers
1384:) CS1 maint: numeric names: authors list (
887:) CS1 maint: numeric names: authors list (
430:are defined topologically, an analogue of
18:
1416:
1350:Algebraic numbers and algebraic functions
1324:
1248:
1207:
547:which are trivial on the embedded field.
91:in 1961–62, it appeared as Volume 144 in
465:This chapter introduces the topological
453:of the field, with the more complicated
418:of finite dimension over a local field.
1463:
1461:
1459:
779:
1615:
1373:
876:
499:for A-fields, describing the units in
1150:
1148:
840:. Bronx, N.Y.: Chelsea Publishing Co.
635:in a simple algebra over an A-field.
7:
1309:"Sur la Théorie du Corps de Classes"
713:of local fields using the theory of
495:The chapter ends with a generalized
1190:IYANAGA et T. TAMAGAWA, S. (1951).
325:Mathematical Association of America
231:diminished the significance of the
203:completions. In this setting, the
14:
851:Hecke, Erich, 1887-1947. (1981).
266:Alongside the desire to consider
1348:Artin, Emil, 1898-1962. (2005).
992:Chevalley, Claude (1933-01-01).
151:Mathematical context and purpose
1639:C. R. (Doklady) Acad. Sci. URSS
1567:Tate, John Torrence Jr (1997).
1405:Japanese Journal of Mathematics
709:'s theorem on the structure of
373:of K over k is induced by the
360:, and without the language of
227:, a series of developments in
1:
855:. New York: Springer-Verlag.
731:Examples of L-functions with
263:during the period 1950–1952.
1494:Basic Number Theory (review)
902:Hasse, Helmut (1930-01-01).
477:both the adele ring and the
378:simplifies the treatment of
83:with particular emphasis on
1552:Minkowski, Hermann (1896).
579:. This gives, for example,
407:Wedderburn's little theorem
1703:
1399:Iyanaga, Shokichi (2006).
607:, and the formula for the
481:group are locally compact;
235:(and, more generally, the
71:is an influential book by
1556:. Leipzig: B. G. Teubner.
1418:10.1007/s11537-006-0502-5
1274:The Annals of Mathematics
1231:Tamagawa, Tsuneo (1951).
1157:The Annals of Mathematics
1119:The Annals of Mathematics
1084:The Annals of Mathematics
1010:10.1515/crll.1933.169.140
963:10.1515/crll.1930.162.145
920:10.1515/crll.1930.162.169
797:10.1007/978-3-662-05978-4
756:10.1007/978-3-642-61945-8
573:meromorphic continuations
457:case postponed to later.
276:global class field theory
1250:10.4099/jjm1924.21.0_197
667:of the field is proved.
649:local class field theory
535:is stated and proved in
380:local class field theory
356:without the language of
302:was first introduced by
280:local class field theory
237:crossed product algebras
16:Book about number theory
1687:Algebraic number theory
591:, and was developed in
268:algebraic number fields
126:, the main theorems of
114:of one variable with a
77:algebraic number theory
1622:: CS1 maint: others (
585:Dedekind zeta-function
375:Frobenius automorphism
369:of A-fields, then any
124:locally compact fields
1326:10.2969/jmsj/00310001
1209:10.2969/jmsj/00310220
836:Hecke, Erich (1970).
661:absolute Galois group
581:analytic continuation
397:The book begins with
141:representation theory
1492:Fernando Q. GouvĂŞa,
1307:Weil, Andre (1951).
787:Weil, André (1973).
746:Weil, André (1974).
653:multiplicative group
577:functional equations
533:Riemann-Roch theorem
319:and Helmut Koch for
317:Mathematical Reviews
110:and of the field of
104:algebraic extensions
89:Princeton University
20:Basic Number Theory
1658:J. Indian Math. Soc
789:Basic Number Theory
748:Basic Number Theory
669:Ramification theory
655:of a field and the
605:ramification theory
539:language, with the
451:separable extension
432:Minkowski's theorem
294:'s suggestion. The
75:, an exposition of
68:Basic Number Theory
21:
1523:10.1007/BF02941019
1053:10.1007/BF02941159
673:abelian extensions
569:Fourier transforms
517:fundamental domain
409:'). Properties of
401:’s formulation of
354:class field theory
229:class field theory
112:rational functions
81:class field theory
1682:Mathematics books
1601:978-0-9502734-2-6
806:978-3-662-05980-7
765:978-3-540-58655-5
665:algebraic closure
623:Chapters X and XI
537:measure-theoretic
523:of an extension.
513:fractional ideals
416:division algebras
362:Galois cohomology
211:) give a natural
209:valuation vectors
137:automorphic forms
102:, meaning finite
64:
63:
59:978-3-540-58655-5
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1447:George Whaples,
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957:(162): 145–154.
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733:Grössencharacter
689:Hasse invariants
436:character groups
367:normal extension
327:and by Koch for
145:algebraic groups
108:rational numbers
106:of the field of
44:Publication date
22:
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657:character group
645:reciprocity law
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541:canonical class
529:
509:
489:Pontryagin dual
463:
447:Tensor products
444:
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387:splitting field
350:simple algebras
342:
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233:cyclic algebras
213:locally compact
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132:simple algebras
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1660:. New Series.
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1641:. New Series.
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675:is developed.
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633:maximal ideals
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571:gives rise to
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278:from those of
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1513:(in German).
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1043:(in German).
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728:distribution.
727:
723:
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711:Galois groups
708:
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695:Third edition
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241:cohomological
238:
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219:
218:global fields
214:
210:
206:
202:
198:
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191:in which the
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100:global fields
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1590:
1585:
1568:
1562:
1553:
1547:
1514:
1510:
1504:
1497:
1488:
1443:
1411:(1): 25–85.
1408:
1404:
1394:
1349:
1343:
1316:
1312:
1302:
1277:
1273:
1267:
1240:
1236:
1226:
1199:
1195:
1185:
1160:
1156:
1122:
1118:
1112:
1087:
1083:
1077:
1047:(1): 46–51.
1044:
1040:
1034:
1001:
997:
987:
954:
950:
944:
911:
907:
897:
852:
846:
837:
831:
821:
815:
788:
782:
747:
720:Theorems of
698:
682:
679:Chapter XIII
642:
626:
618:
602:
599:Chapter VIII
554:
530:
521:discriminant
510:
497:unit theorem
494:
464:
445:
425:
411:Haar measure
403:Wedderburn’s
396:
371:automorphism
345:
343:
329:Zentralblatt
321:Zentralblatt
314:
300:number field
265:
225:World War II
222:
193:real numbers
154:
120:Haar measure
116:finite field
95:
67:
66:
65:
1498:MAA Reviews
1475:(1st ed.),
1319:(1): 1–35.
1243:: 197–215.
715:Weil groups
639:Chapter XII
561:L-functions
551:Chapter VII
455:inseparable
442:Chapter III
333:Shafarevich
296:idèle group
197:completions
38:Mathematics
1676:Categories
1664:: 197–202.
1517:(1): 413.
1481:0823.11001
1473:0176.33601
1280:(2): 348.
774:References
615:Chapter IX
545:adele ring
527:Chapter VI
467:adele ring
461:Chapter IV
422:Chapter II
384:unramified
358:cohomology
245:Hochschild
93:Springer's
73:André Weil
29:André Weil
1618:cite book
1610:665069251
1577:304411725
1539:124096167
1531:1865-8784
1483:(2nd ed.)
1435:123613236
1427:0289-2316
1376:cite book
1335:0025-5645
1259:0075-3432
1218:0025-5645
1069:121475651
1061:1865-8784
1026:115917687
1018:0075-4102
979:116860448
971:0075-4102
936:199546442
928:0075-4102
879:cite book
707:Ĺ afareviÄŤ
507:Chapter V
501:valuation
393:Chapter I
311:Reception
304:Chevalley
284:Chevalley
272:Chevalley
173:Chevalley
85:valuation
1645:: 15–16.
1573:ProQuest
1368:62741519
740:Editions
726:Herbrand
629:groupoid
428:lattices
340:Contents
249:Nakayama
189:Tamagawa
1452:0234930
1294:1969783
1177:1969801
1139:1970190
1104:1969863
871:7576150
663:of the
659:of the
583:of the
503:terms.
181:Iwasawa
161:measure
157:Hecke's
1608:
1598:
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1529:
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1471:
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1425:
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1216:
1175:
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1067:
1059:
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1016:
977:
969:
934:
926:
869:
859:
803:
762:
259:, and
223:After
205:adeles
201:p-adic
187:, and
165:Hensel
128:adelic
25:Author
1592:Union
1535:S2CID
1431:S2CID
1290:JSTOR
1173:JSTOR
1135:JSTOR
1100:JSTOR
1065:S2CID
1022:S2CID
975:S2CID
932:S2CID
685:idèle
609:genus
589:Artin
565:idèle
479:idèle
471:idèle
298:of a
292:Hasse
290:, at
288:idèle
257:Artin
177:Artin
169:Hasse
35:Genre
1624:link
1606:OCLC
1596:ISBN
1527:ISSN
1423:ISSN
1386:link
1382:link
1364:OCLC
1354:ISBN
1331:ISSN
1255:ISSN
1214:ISSN
1057:ISSN
1014:ISSN
1002:1933
967:ISSN
955:1930
924:ISSN
912:1930
889:link
885:link
867:OCLC
857:ISBN
801:ISBN
760:ISBN
722:Tate
671:for
643:The
575:and
559:and
557:zeta
555:The
469:and
399:Witt
352:and
261:Tate
253:Weil
247:and
207:(or
185:Tate
79:and
54:ISBN
48:1974
1519:doi
1477:Zbl
1469:Zbl
1413:doi
1321:doi
1282:doi
1245:doi
1204:doi
1165:doi
1127:doi
1092:doi
1049:doi
1006:doi
959:doi
916:doi
793:doi
752:doi
647:of
631:of
143:of
122:on
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