658:
507:
549:
434:
825:
211:
357:
324:
302:
240:
168:
142:
120:
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554:
666:
of all CM fields is similar to a CM field except that it has infinite degree. It is a quadratic extension of the union of all totally real fields
896:
214:
759:
is an order in a CM-field. Conversely any CM field arises like this from some simple complex abelian variety, unique up to isogeny.
928:
450:
923:
712:) has a center generated by an element of order 2 (complex conjugation) and the quotient by its center is the group Gal(
376:
389:
933:
862:, Publications of the Mathematical Society of Japan, vol. 11, Princeton, N.J.: Princeton University Press
516:
401:
871:, Publications of the Mathematical Society of Japan, vol. 6, Tokyo: The Mathematical Society of Japan,
93:
765:
43:
671:
218:
762:
One example of a totally imaginary field which is not CM is the number field defined by the polynomial
192:
510:
441:
279:
70:
340:
307:
285:
223:
151:
125:
103:
445:
85:
892:
902:
847:
396:
333:
is a CM-field if and only if it has a "units defect", i.e. if it contains a proper subfield
876:
906:
888:
872:
851:
253:
243:
917:
437:
838:
Remak, Robert (1954), "Über algebraische Zahlkörper mit schwachem
Einheitsdefekt",
39:
653:{\displaystyle \zeta _{n}^{2}+\zeta _{n}^{-2}-2=(\zeta _{n}-\zeta _{n}^{-1})^{2}.}
869:
Complex multiplication of abelian varieties and its applications to number theory
304:
induces an automorphism on the field which is independent of its embedding into
31:
684:) is generated (as a closed subgroup) by all elements of order 2 in Gal(
17:
392:, for which the totally real subfield is just the field of rationals.
860:
Introduction to the arithmetic theory of automorphic functions
388:
The simplest, and motivating, example of a CM-field is an
502:{\displaystyle \mathbb {Q} (\zeta _{n}+\zeta _{n}^{-1}).}
395:
One of the most important examples of a CM-field is the
326:. In the notation given, it must change the sign of β.
768:
557:
519:
453:
404:
343:
310:
288:
256:
226:
195:
154:
128:
106:
189:
by a single square root of an element, say β =
551:is obtained from it by adjoining a square root of
270:into the real number field, σ(α) < 0.
42:, so named for a close connection to the theory of
819:
652:
543:
501:
428:
351:
318:
296:
262:
234:
205:
162:
136:
114:
704:) is a subgroup of index 2. The Galois group Gal(
54:
8:
727:is a complex abelian variety of dimension
799:
786:
773:
767:
641:
628:
623:
610:
585:
580:
567:
562:
556:
532:
521:
520:
518:
484:
479:
466:
455:
454:
452:
417:
406:
405:
403:
345:
344:
342:
312:
311:
309:
290:
289:
287:
255:
228:
227:
225:
196:
194:
156:
155:
153:
130:
129:
127:
108:
107:
105:
53:The abbreviation "CM" was introduced by (
867:Shimura, Goro; Taniyama, Yutaka (1961),
544:{\displaystyle \mathbb {Q} (\zeta _{n})}
436:, which is generated by a primitive nth
429:{\displaystyle \mathbb {Q} (\zeta _{n})}
820:{\displaystyle x^{4}+x^{3}-x^{2}-x+1}
364:
7:
173:In other words, there is a subfield
375:mentioned above. This follows from
278:One feature of a CM-field is that
250:, so that for each embedding σ of
25:
885:Introduction to Cyclotomic fields
509:The latter is the fixed field of
206:{\displaystyle {\sqrt {\alpha }}}
883:Washington, Lawrence C. (1996).
371:is the totally real subfield of
144:, but there is no embedding of
638:
603:
538:
525:
493:
459:
423:
410:
337:whose unit group has the same
246:. For this α should be chosen
1:
440:. It is a totally imaginary
352:{\displaystyle \mathbb {Z} }
319:{\displaystyle \mathbb {C} }
297:{\displaystyle \mathbb {C} }
235:{\displaystyle \mathbb {Q} }
163:{\displaystyle \mathbb {R} }
137:{\displaystyle \mathbb {R} }
115:{\displaystyle \mathbb {C} }
27:Complex multiplication field
731:, then any abelian algebra
242:has all its roots non-real
96:. I.e., every embedding of
55:Shimura & Taniyama 1961
950:
887:(2nd ed.). New York:
390:imaginary quadratic field
213:, in such a way that the
69:is a CM-field if it is a
377:Dirichlet's unit theorem
46:. Another name used is
38:is a particular type of
929:Algebraic number theory
858:Shimura, Goro (1971),
840:Compositio Mathematica
821:
654:
545:
503:
430:
353:
320:
298:
264:
236:
207:
164:
138:
116:
44:complex multiplication
822:
672:absolute Galois group
655:
546:
504:
431:
354:
321:
299:
265:
237:
219:rational number field
208:
165:
139:
122:lies entirely within
117:
80:where the base field
766:
735:of endomorphisms of
555:
517:
451:
402:
341:
308:
286:
254:
224:
193:
152:
126:
104:
924:Field (mathematics)
636:
593:
572:
511:complex conjugation
492:
442:quadratic extension
280:complex conjugation
71:quadratic extension
817:
747:. If it has rank 2
739:has rank at most 2
650:
619:
576:
558:
541:
499:
475:
446:totally real field
426:
349:
316:
294:
260:
232:
215:minimal polynomial
203:
185:is generated over
160:
134:
112:
359:-rank as that of
263:{\displaystyle F}
201:
94:totally imaginary
61:Formal definition
16:(Redirected from
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397:cyclotomic field
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248:totally negative
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934:Complex numbers
914:
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889:Springer-Verlag
882:
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857:
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755:is simple then
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329:A number field
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244:complex numbers
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65:A number field
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28:
23:
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217:of β over the
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2:
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898:0-387-94762-0
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842:(in German),
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439:
438:root of unity
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147:
99:
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83:
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72:
68:
60:
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56:
51:
49:
45:
41:
37:
33:
19:
884:
868:
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843:
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748:
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724:
717:
713:
709:
705:
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696:
691:
686:
681:
676:
667:
663:
372:
368:
367:). In fact,
360:
334:
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328:
277:
247:
186:
182:
178:
174:
172:
145:
97:
89:
86:totally real
81:
77:
73:
66:
64:
52:
47:
40:number field
35:
29:
694:), and Gal(
32:mathematics
918:Categories
907:0966.11047
852:0055.26805
832:References
662:The union
365:Remak 1954
274:Properties
181:such that
846:: 35–80,
806:−
793:−
630:−
621:ζ
617:−
608:ζ
595:−
587:−
578:ζ
560:ζ
530:ζ
486:−
477:ζ
464:ζ
415:ζ
199:α
383:Examples
36:CM-field
18:CM field
877:0125113
444:of the
48:J-field
905:
895:
875:
850:
670:. The
513:, and
743:over
148:into
100:into
893:ISBN
751:and
674:Gal(
88:but
34:, a
903:Zbl
848:Zbl
723:If
282:on
177:of
92:is
84:is
57:).
30:In
920::
901:.
891:.
873:MR
844:12
720:).
379:.
170:.
50:.
909:.
827:.
815:1
812:+
809:x
801:2
797:x
788:3
784:x
780:+
775:4
771:x
757:F
753:V
749:n
745:Z
741:n
737:V
733:F
729:n
725:V
718:Q
716:/
714:Q
710:Q
708:/
706:Q
702:Q
700:/
697:Q
692:Q
690:/
687:Q
682:Q
680:/
677:Q
668:Q
664:Q
648:.
643:2
639:)
633:1
625:n
612:n
604:(
601:=
598:2
590:2
582:n
574:+
569:2
564:n
539:)
534:n
526:(
522:Q
497:.
494:)
489:1
481:n
473:+
468:n
460:(
456:Q
424:)
419:n
411:(
407:Q
373:K
369:F
363:(
361:K
346:Z
335:F
331:K
313:C
291:C
258:F
229:Q
187:F
183:K
179:K
175:F
157:R
146:K
131:R
109:C
98:F
90:K
82:F
78:F
76:/
74:K
67:K
20:)
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