28:
403: ≠2, its absolute Galois group is generated by + 3 elements and has an explicit description by generators and relations. This is a result of Uwe Jannsen and Kay Wingberg. Some results are known in the case
270:
881:
519:
531:
Every profinite group occurs as a Galois group of some Galois extension, however not every profinite group occurs as an absolute Galois group. For example, the
493:
207:
325:
The absolute Galois group of the field of rational functions with complex coefficients is free (as a profinite group). This result is due to
369:
970:(1995), "Étale Galois covers of affine smooth curves. The geometric case of a conjecture of Shafarevich. On Abhyankar's conjecture",
942:
771:
17:
535:
asserts that the only finite absolute Galois groups are either trivial or of order 2, that is only two isomorphism classes.
543:
330:
842:
1051:
539:
862:
972:
890:
838:
1029:
Harpaz, Yonatan; Wittenberg, Olivier (2023), "The Massey vanishing conjecture for number fields",
1067:
286:
498:
1001:
981:
952:
899:
850:
819:
762:, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, vol. 11 (3rd ed.),
751:
663:
441:
960:
781:
8:
150:
66:
985:
903:
1005:
915:
858:
547:
478:
436:
417:
Another case in which the absolute Galois group has been determined is for the largest
112:
926:
449:(maps on surfaces), enabling us to "see" the Galois theory of algebraic number fields.
1009:
938:
919:
767:
551:
532:
457:
196:
135:
97:
89:
472:
177:
is a cyclic group of two elements (complex conjugation and the identity map), since
989:
956:
907:
805:
777:
384:
997:
948:
846:
815:
763:
747:
432:
116:
1016:
Mináč, Ján; Tân, Nguyên Duy (2016), "Triple Massey products and Galois Theory",
826:
361:
1061:
810:
388:
326:
278:
265:{\displaystyle {\hat {\mathbf {Z} }}=\varprojlim \mathbf {Z} /n\mathbf {Z} .}
127:
27:
446:
418:
373:
189:
158:
73:
36:
32:
967:
365:
174:
52:
993:
911:
170:
The absolute Galois group of an algebraically closed field is trivial.
1047:
835:
Recent developments in the inverse Galois problem (Seattle, WA, 1993)
829:(1995), "Fundamental groups and embedding problems in characteristic
431:
No direct description is known for the absolute Galois group of the
788:
Haran, Dan; Jarden, Moshe (2000), "The absolute Galois group of
738:
Douady, Adrien (1964), "DĂ©termination d'un groupe de Galois",
109:
1046:, Cambridge studies in advanced mathematics, vol. 117,
439:
that the absolute Galois group has a faithful action on the
96:. Alternatively it is the group of all automorphisms of the
925:
685:
650:
638:
865:
501:
481:
210:
475:and Nguyên Duy Tân's conjecture about vanishing of
39:of order 2 generated by complex conjugation, since
875:
740:Comptes Rendus de l'Académie des Sciences de Paris
513:
487:
264:
542:can be realized as an absolute Galois group of a
1059:
1028:
856:
627:
344:a variable. Then the absolute Galois group of
356:) is free of rank equal to the cardinality of
935:Grundlehren der Mathematischen Wissenschaften
289:Fr is a canonical (topological) generator of
929:; Schmidt, Alexander; Wingberg, Kay (2000),
108:. The absolute Galois group is well-defined
837:, Contemporary Mathematics, vol. 186,
787:
757:
616:
421:subfield of the field of algebraic numbers.
937:, vol. 323, Berlin: Springer-Verlag,
464:asserts that the absolute Galois group of
859:"Die Struktur der absoluten Galoisgruppe
809:
758:Fried, Michael D.; Jarden, Moshe (2008),
1041:
1018:Journal of European Mathematical Society
1015:
825:
594:
571:
26:
525:
14:
1060:
737:
706:Harpaz & Wittenberg (2023) pp.1,41
700:
691:
583:
718:
686:Neukirch, Schmidt & Wingberg 2000
651:Neukirch, Schmidt & Wingberg 2000
639:Neukirch, Schmidt & Wingberg 2000
340:be an algebraically closed field and
1044:Galois Groups and Fundamental Groups
857:Jannsen, Uwe; Wingberg, Kay (1982),
724:Fried & Jarden (2008) pp.208,545
709:
471:An interesting problem is to settle
966:
868:
605:
24:
25:
1079:
697:Mináč & Tân (2016) pp.255,284
544:pseudo algebraically closed field
173:The absolute Galois group of the
31:The absolute Galois group of the
435:. In this case, it follows from
255:
242:
215:
876:{\displaystyle {\mathfrak {p}}}
679:
368:, and was also proved later by
188:The absolute Galois group of a
798:Pacific Journal of Mathematics
715:Fried & Jarden (2008) p.12
656:
644:
632:
621:
610:
599:
588:
577:
565:
460:of the rational numbers. Then
219:
195:is isomorphic to the group of
13:
1:
843:American Mathematical Society
558:
316:is the number of elements in
181:is the separable closure of
43:is the separable closure of
7:
931:Cohomology of Number Fields
628:Jannsen & Wingberg 1982
425:
407:= 2, but the structure for
331:Riemann's existence theorem
164:
10:
1084:
1052:Cambridge University Press
731:
540:projective profinite group
468:is a free profinite group.
1031:Duke Mathematical Journal
1042:Szamuely, Tamás (2009),
973:Inventiones Mathematicae
891:Inventiones Mathematicae
839:Providence, Rhode Island
811:10.2140/pjm.2000.196.445
546:. This result is due to
462:Shafarevich's conjecture
376:using algebraic methods.
360:. This result is due to
617:Haran & Jarden 2000
514:{\displaystyle n\geq 3}
329:and has its origins in
277:(For the notation, see
877:
533:Artin–Schreier theorem
515:
495:- Massey products for
489:
287:Frobenius automorphism
266:
145:. This holds e.g. for
48:
883:-adischer Zahlkörper"
878:
516:
490:
267:
57:absolute Galois group
30:
18:Absolute galois group
863:
845:, pp. 353–369,
526:Some general results
499:
479:
336:More generally, let
208:
986:1995InMat.120..555P
904:1982InMat..70...71J
185:and = 2.
151:characteristic zero
47:and = 2.
994:10.1007/bf01241142
912:10.1007/bf01393199
873:
548:Alexander Lubotzky
511:
485:
296:. (Recall that Fr(
262:
236:
197:profinite integers
134:is the same as an
113:inner automorphism
49:
944:978-3-540-66671-4
773:978-3-540-77269-9
552:Lou van den Dries
488:{\displaystyle n}
458:abelian extension
442:dessins d'enfants
229:
222:
136:algebraic closure
98:algebraic closure
90:separable closure
16:(Redirected from
1075:
1054:
1038:
1025:
1012:
963:
927:Neukirch, JĂĽrgen
922:
887:
882:
880:
879:
874:
872:
871:
853:
822:
813:
784:
760:Field arithmetic
754:
725:
722:
716:
713:
707:
704:
698:
695:
689:
683:
677:
676:
674:
673:
668:
660:
654:
648:
642:
641:, theorem 7.5.10
636:
630:
625:
619:
614:
608:
603:
597:
592:
586:
581:
575:
569:
520:
518:
517:
512:
494:
492:
491:
486:
433:rational numbers
385:finite extension
271:
269:
268:
263:
258:
250:
245:
237:
224:
223:
218:
213:
21:
1083:
1082:
1078:
1077:
1076:
1074:
1073:
1072:
1058:
1057:
945:
885:
867:
866:
864:
861:
860:
827:Harbater, David
774:
764:Springer-Verlag
734:
729:
728:
723:
719:
714:
710:
705:
701:
696:
692:
684:
680:
671:
669:
666:
662:
661:
657:
649:
645:
637:
633:
626:
622:
615:
611:
604:
600:
593:
589:
582:
578:
570:
566:
561:
528:
500:
497:
496:
480:
477:
476:
456:be the maximal
437:Belyi's theorem
428:
413:
398:
294:
254:
246:
241:
228:
214:
212:
211:
209:
206:
205:
167:
117:profinite group
63:
23:
22:
15:
12:
11:
5:
1081:
1071:
1070:
1056:
1055:
1039:
1026:
1013:
980:(3): 555–578,
964:
943:
923:
870:
854:
823:
804:(2): 445–459,
785:
772:
755:
733:
730:
727:
726:
717:
708:
699:
690:
678:
655:
643:
631:
620:
609:
598:
587:
576:
563:
562:
560:
557:
556:
555:
536:
527:
524:
523:
522:
510:
507:
504:
484:
469:
450:
427:
424:
423:
422:
415:
411:
394:
389:p-adic numbers
377:
362:David Harbater
334:
322:
321:
292:
275:
274:
273:
272:
261:
257:
253:
249:
244:
240:
235:
232:
227:
221:
217:
200:
199:
186:
171:
166:
163:
61:
9:
6:
4:
3:
2:
1080:
1069:
1068:Galois theory
1066:
1065:
1063:
1053:
1049:
1045:
1040:
1036:
1032:
1027:
1023:
1019:
1014:
1011:
1007:
1003:
999:
995:
991:
987:
983:
979:
975:
974:
969:
965:
962:
958:
954:
950:
946:
940:
936:
932:
928:
924:
921:
917:
913:
909:
905:
901:
897:
893:
892:
884:
855:
852:
848:
844:
840:
836:
832:
828:
824:
821:
817:
812:
807:
803:
799:
795:
791:
786:
783:
779:
775:
769:
765:
761:
756:
753:
749:
746:: 5305–5308,
745:
741:
736:
735:
721:
712:
703:
694:
687:
682:
665:
659:
652:
647:
640:
635:
629:
624:
618:
613:
607:
602:
596:
595:Harbater 1995
591:
585:
580:
573:
572:Szamuely 2009
568:
564:
553:
549:
545:
541:
537:
534:
530:
529:
508:
505:
502:
482:
474:
470:
467:
463:
459:
455:
451:
448:
444:
443:
438:
434:
430:
429:
420:
416:
414:is not known.
410:
406:
402:
397:
393:
390:
386:
382:
378:
375:
371:
367:
363:
359:
355:
351:
348: =
347:
343:
339:
335:
332:
328:
327:Adrien Douady
324:
323:
319:
315:
311:
307:
303:
299:
295:
288:
284:
283:
282:
280:
279:Inverse limit
259:
251:
247:
238:
233:
230:
225:
204:
203:
202:
201:
198:
194:
191:
187:
184:
180:
176:
172:
169:
168:
162:
160:
156:
152:
148:
144:
140:
137:
133:
129:
128:perfect field
125:
120:
118:
114:
111:
107:
103:
99:
95:
91:
87:
83:
79:
75:
71:
68:
64:
58:
54:
46:
42:
38:
34:
29:
19:
1043:
1034:
1030:
1024:(1): 255–284
1021:
1017:
977:
971:
968:Pop, Florian
934:
930:
895:
889:
834:
830:
801:
797:
793:
789:
759:
743:
739:
720:
711:
702:
693:
681:
670:. Retrieved
658:
646:
634:
623:
612:
601:
590:
579:
567:
465:
461:
453:
447:Grothendieck
440:
419:totally real
408:
404:
400:
395:
391:
380:
374:Moshe Jarden
357:
353:
349:
345:
341:
337:
317:
313:
309:
305:
301:
297:
290:
276:
192:
190:finite field
182:
178:
175:real numbers
159:finite field
154:
146:
142:
138:
131:
123:
121:
105:
101:
93:
85:
81:
77:
74:Galois group
69:
59:
56:
50:
44:
40:
37:cyclic group
33:real numbers
584:Douady 1964
366:Florian Pop
53:mathematics
961:0948.11001
782:1145.12001
672:2019-09-04
559:References
115:. It is a
1048:Cambridge
1037:(1): 1–41
1010:128157587
920:119378923
898:: 71–78,
688:, p. 449.
506:≥
473:Ján Mináč
370:Dan Haran
239:
234:←
220:^
104:that fix
1062:Category
653:, §VII.5
606:Pop 1995
574:, p. 14.
426:Problems
312:, where
304:for all
165:Examples
84:, where
1002:1334484
982:Bibcode
953:1737196
900:Bibcode
851:1352282
820:1800587
752:0162796
732:Sources
387:of the
72:is the
1008:
1000:
959:
951:
941:
918:
849:
818:
780:
770:
750:
538:Every
399:. For
122:(When
55:, the
1006:S2CID
916:S2CID
886:(PDF)
667:(PDF)
664:"qtr"
383:be a
153:, or
126:is a
110:up to
88:is a
80:over
67:field
65:of a
35:is a
939:ISBN
796:)",
768:ISBN
550:and
452:Let
379:Let
372:and
364:and
300:) =
285:The
1035:172
990:doi
978:120
957:Zbl
908:doi
833:",
806:doi
802:196
778:Zbl
744:258
445:of
308:in
281:.)
231:lim
161:.)
149:of
141:of
100:of
92:of
76:of
51:In
1064::
1050::
1033:,
1022:19
1020:,
1004:,
998:MR
996:,
988:,
976:,
955:,
949:MR
947:,
933:,
914:,
906:,
896:70
894:,
888:,
847:MR
841::
816:MR
814:,
800:,
776:,
766:,
748:MR
742:,
320:.)
157:a
130:,
119:.
992::
984::
910::
902::
869:p
831:p
808::
794:x
792:(
790:C
675:.
554:.
521:.
509:3
503:n
483:n
466:K
454:K
412:2
409:Q
405:p
401:p
396:p
392:Q
381:K
358:C
354:x
352:(
350:C
346:K
342:x
338:C
333:.
318:K
314:q
310:K
306:x
302:x
298:x
293:K
291:G
260:.
256:Z
252:n
248:/
243:Z
226:=
216:Z
193:K
183:R
179:C
155:K
147:K
143:K
139:K
132:K
124:K
106:K
102:K
94:K
86:K
82:K
78:K
70:K
62:K
60:G
45:R
41:C
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
