25:
320:
403:
432:
349:
718:
681:. In fact, M yields a duality (contravariant equivalence) between the category of compact connected Riemann surfaces (with non-constant
89:
61:
42:
68:
1079:
546:
645:. In fact, the function field yields a duality between the category of regular projective irreducible algebraic curves (with
75:
1125:
108:
57:
649:
721:.) In the context of this analogy, both number fields and function fields over finite fields are usually called "
1084:
46:
646:
1147:
276:
354:
451:
82:
1059:
574:
819:
573:
varieties may have the same function field!) Assigning to each variety its function field yields a
893:
741:
35:
752:
706:
447:
237:
175:
733:
570:
664:
630:
590:
156:
408:
325:
8:
566:
233:
138:
1089:
498:
434:). We see that the degree of an algebraic function field is not a well-defined notion.
248:
203:
540:
1121:
780:
772:
463:
582:
1030:
There are natural bijective correspondences between the set of valuation rings of
1115:
1094:
748:
668:
223:
145:
633:
sense) is especially important, since every function field of one variable over
737:
675:
638:
1141:
690:
252:
729:
722:
717:, and these counterparts are frequently easier to prove. (For example, see
714:
710:
682:
609:
122:
569:
if and only if their function fields are isomorphic. (But note that non-
924:
652:
as morphisms) and the category of function fields of one variable over
728:
The study of function fields over a finite field has applications in
24:
1055:
577:(contravariant equivalence) between the category of varieties over
541:
Function fields arising from varieties, curves and
Riemann surfaces
16:
Finitely generated extension field of positive transcendence degree
846:
585:
as morphisms) and the category of algebraic function fields over
641:(i.e. non-singular) projective irreducible algebraic curve over
713:
have a counterpart on function fields of one variable over a
1113:
689:. A similar correspondence exists between compact connected
685:
maps as morphisms) and function fields of one variable over
740:
over a finite field (an important mathematical tool for
719:
Analogue for irreducible polynomials over a finite field
589:. (The varieties considered here are to be taken in the
629: = 1 (irreducible algebraic curves in the
411:
357:
328:
279:
637:arises as the function field of a uniquely defined
49:. Unsourced material may be challenged and removed.
700:
426:
397:
343:
314:
1117:Topics in the Theory of Algebraic Function Fields
818:Key tools to study algebraic function fields are
269:). This is a function field of one variable over
1139:
166:. Equivalently, an algebraic function field of
674:is a function field of one variable over the
1114:Gabriel Daniel & Villa Salvador (2007).
787:. These elements form a field, known as the
833:of one variable, we define the notion of a
802:) is a function field of one variable over
693:and function fields in one variable over
109:Learn how and when to remove this message
1046:, and the set of discrete valuations of
813:
736:. For example, the function field of an
751:play also an important role in solving
1140:
1080:function field of an algebraic variety
547:function field of an algebraic variety
758:
437:
315:{\displaystyle k(X)({\sqrt {X^{3}}})}
1054:. These sets can be given a natural
398:{\displaystyle k(Y)({\sqrt{Y^{2}}})}
47:adding citations to reliable sources
18:
820:absolute values, valuations, places
763:Given any algebraic function field
709:states that almost all theorems on
529:, then there are no morphisms from
442:The algebraic function fields over
13:
896:and its maximal ideal is called a
825:Given an algebraic function field
747:Function fields over the field of
744:) is an algebraic function field.
557:is an algebraic function field of
14:
1159:
791:of the algebraic function field.
597:-rational points, like the curve
892:. Each such valuation ring is a
23:
701:Number fields and finite fields
34:needs additional citations for
1107:
1085:function field (scheme theory)
943:(x) = ∞ iff
593:sense; they need not have any
421:
415:
392:
370:
367:
361:
338:
332:
309:
292:
289:
283:
1:
1100:
806:; its field of constants is
273:; it can also be written as
7:
1073:
10:
1164:
1019:) = 0 for all
497:. All these morphisms are
217:
58:"Algebraic function field"
505:is a function field over
868:, and such that for any
144:is a finitely generated
127:algebraic function field
1038:, the set of places of
894:discrete valuation ring
822:and their completions.
753:inverse Galois problems
742:public key cryptography
667:defined on a connected
567:birationally equivalent
517:is a function field in
860:and is different from
771:, we can consider the
734:error correcting codes
707:function field analogy
583:dominant rational maps
428:
399:
345:
316:
238:irreducible polynomial
222:As an example, in the
176:finite field extension
129:(often abbreviated as
1060:Zariski–Riemann space
1023: ∈
1007: ∈
939:∪{∞} such that
888: ∈
880: ∈
814:Valuations and places
665:meromorphic functions
429:
400:
346:
317:
1027: \ {0}.
565:. Two varieties are
454:from function field
427:{\displaystyle k(Y)}
409:
405:(with degree 3 over
355:
344:{\displaystyle k(X)}
326:
322:(with degree 2 over
277:
174:may be defined as a
157:transcendence degree
43:improve this article
1148:Field (mathematics)
1090:algebraic function
947: = 0,
913:discrete valuation
789:field of constants
759:Field of constants
464:ring homomorphisms
438:Category structure
424:
395:
341:
312:
249:field of fractions
204:rational functions
608:defined over the
390:
307:
236:generated by the
137:variables over a
119:
118:
111:
93:
1155:
1132:
1131:
1111:
963:) +
749:rational numbers
621:
607:
433:
431:
430:
425:
404:
402:
401:
396:
391:
389:
384:
383:
374:
350:
348:
347:
342:
321:
319:
318:
313:
308:
306:
305:
296:
260:
231:
114:
107:
103:
100:
94:
92:
51:
27:
19:
1163:
1162:
1158:
1157:
1156:
1154:
1153:
1152:
1138:
1137:
1136:
1135:
1128:
1112:
1108:
1103:
1095:Drinfeld module
1076:
1058:structure: the
979: +
816:
775:of elements of
761:
703:
676:complex numbers
669:Riemann surface
613:
612:, that is with
598:
561:variables over
543:
521:variables, and
513:variables, and
440:
410:
407:
406:
385:
379:
375:
373:
356:
353:
352:
327:
324:
323:
301:
297:
295:
278:
275:
274:
265: −
258:
243: −
229:
224:polynomial ring
220:
210:variables over
201:
192:
170:variables over
146:field extension
115:
104:
98:
95:
52:
50:
40:
28:
17:
12:
11:
5:
1161:
1151:
1150:
1134:
1133:
1126:
1105:
1104:
1102:
1099:
1098:
1097:
1092:
1087:
1082:
1075:
1072:
856:that contains
835:valuation ring
815:
812:
794:For instance,
760:
757:
738:elliptic curve
702:
699:
691:Klein surfaces
542:
539:
439:
436:
423:
420:
417:
414:
394:
388:
382:
378:
372:
369:
366:
363:
360:
340:
337:
334:
331:
311:
304:
300:
294:
291:
288:
285:
282:
219:
216:
197:
190:
131:function field
117:
116:
31:
29:
22:
15:
9:
6:
4:
3:
2:
1160:
1149:
1146:
1145:
1143:
1129:
1127:9780817645151
1123:
1119:
1118:
1110:
1106:
1096:
1093:
1091:
1088:
1086:
1083:
1081:
1078:
1077:
1071:
1069:
1065:
1061:
1057:
1053:
1049:
1045:
1041:
1037:
1033:
1028:
1026:
1022:
1018:
1014:
1010:
1006:
1002:
998:
994:
990:
986:
982:
978:
974:
970:
966:
962:
958:
954:
950:
946:
942:
938:
934:
930:
926:
922:
918:
914:
909:
907:
903:
899:
895:
891:
887:
883:
879:
875:
871:
867:
863:
859:
855:
851:
848:
844:
840:
836:
832:
828:
823:
821:
811:
809:
805:
801:
797:
792:
790:
786:
782:
778:
774:
770:
766:
756:
754:
750:
745:
743:
739:
735:
731:
726:
724:
723:global fields
720:
716:
712:
711:number fields
708:
698:
696:
692:
688:
684:
680:
677:
673:
670:
666:
662:
657:
655:
651:
648:
644:
640:
636:
632:
628:
623:
620:
616:
611:
605:
601:
596:
592:
588:
584:
580:
576:
572:
568:
564:
560:
556:
552:
549:of dimension
548:
538:
536:
532:
528:
524:
520:
516:
512:
508:
504:
500:
496:
492:
488:
484:
480:
476:
472:
469: :
468:
465:
461:
457:
453:
449:
445:
435:
418:
412:
386:
380:
376:
364:
358:
335:
329:
302:
298:
286:
280:
272:
268:
264:
257:
254:
253:quotient ring
250:
247:and form the
246:
242:
239:
235:
232:consider the
228:
225:
215:
213:
209:
205:
200:
196:
189:
185:
181:
178:of the field
177:
173:
169:
165:
161:
158:
154:
150:
147:
143:
140:
136:
132:
128:
124:
113:
110:
102:
99:December 2021
91:
88:
84:
81:
77:
74:
70:
67:
63:
60: –
59:
55:
54:Find sources:
48:
44:
38:
37:
32:This article
30:
26:
21:
20:
1120:. Springer.
1116:
1109:
1067:
1063:
1051:
1047:
1043:
1039:
1035:
1031:
1029:
1024:
1020:
1016:
1012:
1008:
1004:
1000:
996:
992:
988:
984:
980:
976:
972:
968:
964:
960:
956:
952:
948:
944:
940:
936:
932:
928:
920:
916:
912:
910:
905:
901:
897:
889:
885:
881:
877:
873:
869:
865:
861:
857:
853:
849:
845:: this is a
842:
838:
834:
830:
826:
824:
817:
807:
803:
799:
795:
793:
788:
784:
776:
768:
764:
762:
746:
730:cryptography
727:
715:finite field
704:
694:
686:
678:
671:
660:
659:The field M(
658:
653:
650:regular maps
642:
634:
626:
624:
618:
614:
603:
599:
594:
586:
578:
562:
558:
554:
550:
544:
534:
530:
526:
522:
518:
514:
510:
506:
502:
494:
490:
486:
482:
478:
474:
470:
466:
459:
455:
443:
441:
270:
266:
262:
255:
244:
240:
226:
221:
211:
207:
198:
194:
187:
183:
179:
171:
167:
163:
159:
152:
148:
141:
134:
130:
126:
120:
105:
96:
86:
79:
72:
65:
53:
41:Please help
36:verification
33:
1056:topological
999:)) for all
683:holomorphic
123:mathematics
1101:References
925:surjective
779:which are
571:isomorphic
155:which has
69:newspapers
927:function
781:algebraic
625:The case
499:injective
452:morphisms
1142:Category
1074:See also
1003:,
983:) ≥ min(
931: :
876:we have
647:dominant
489:for all
462:are the
448:category
351:) or as
847:subring
639:regular
606:+ 1 = 0
575:duality
446:form a
259:
251:of the
230:
218:Example
83:scholar
1124:
1011:, and
971:) and
631:scheme
591:scheme
581:(with
450:; the
85:
78:
71:
64:
56:
923:is a
898:place
783:over
767:over
663:) of
610:reals
553:over
525:>
501:. If
477:with
234:ideal
202:) of
193:,...,
162:over
139:field
133:) of
125:, an
90:JSTOR
76:books
1122:ISBN
955:) =
864:and
732:and
705:The
545:The
485:) =
62:news
1062:of
915:of
900:of
884:or
872:in
852:of
837:of
773:set
725:".
622:.)
533:to
509:of
493:in
458:to
206:in
121:In
45:by
1144::
1070:.
991:),
953:xy
935:→
911:A
908:.
810:.
755:.
697:.
656:.
617:=
602:+
537:.
473:→
261:/(
214:.
182:=
1130:.
1068:k
1066:/
1064:K
1052:k
1050:/
1048:K
1044:k
1042:/
1040:K
1036:k
1034:/
1032:K
1025:k
1021:a
1017:a
1015:(
1013:v
1009:K
1005:y
1001:x
997:y
995:(
993:v
989:x
987:(
985:v
981:y
977:x
975:(
973:v
969:y
967:(
965:v
961:x
959:(
957:v
951:(
949:v
945:x
941:v
937:Z
933:K
929:v
921:k
919:/
917:K
906:k
904:/
902:K
890:O
886:x
882:O
878:x
874:K
870:x
866:K
862:k
858:k
854:K
850:O
843:k
841:/
839:K
831:k
829:/
827:K
808:C
804:R
800:x
798:(
796:C
785:k
777:K
769:k
765:K
695:R
687:C
679:C
672:X
661:X
654:k
643:k
635:k
627:n
619:R
615:k
604:Y
600:X
595:k
587:k
579:k
563:k
559:n
555:k
551:n
535:L
531:K
527:m
523:n
519:m
515:L
511:n
507:k
503:K
495:k
491:a
487:a
483:a
481:(
479:f
475:L
471:K
467:f
460:L
456:K
444:k
422:)
419:Y
416:(
413:k
393:)
387:3
381:2
377:Y
371:(
368:)
365:Y
362:(
359:k
339:)
336:X
333:(
330:k
310:)
303:3
299:X
293:(
290:)
287:X
284:(
281:k
271:k
267:X
263:Y
256:k
245:X
241:Y
227:k
212:k
208:n
199:n
195:x
191:1
188:x
186:(
184:k
180:K
172:k
168:n
164:k
160:n
153:k
151:/
149:K
142:k
135:n
112:)
106:(
101:)
97:(
87:·
80:·
73:·
66:·
39:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
