514:
368:
690:
a
Pythagorean field satisfies many of Hilbert's axioms, such as the incidence axioms, the congruence axioms and the axioms of parallels. However, in general this geometry need not satisfy all Hilbert's axioms unless the field
695:
has extra properties: for example, if the field is also ordered then the geometry will satisfy
Hilbert's ordering axioms, and if the field is also complete the geometry will satisfy Hilbert's completeness axiom.
461:
87:
735:
183:
760:
287:
237:
1387:
107:
762:
can be used to construct non-archimedean geometries that satisfy many of
Hilbert's axioms but not his axiom of completeness. Dehn used such a field to construct two
668:
403:
498:
688:
423:
272:
151:
127:
50:
774:
respectively, in which there are many lines though a point not intersecting a given line but where the sum of the angles of a triangle is at least π.
1334:
1258:
1358:
1429:
1399:
1370:
1295:
1124:
553:
1322:
1147:
1326:
1287:
1116:
700:
535:
524:
1455:
213:
in which all non-negative elements are squares) is an ordered
Pythagorean field, but the converse does not hold. A
1421:
214:
428:
55:
1194:(2010), "Old and new results in the foundations of elementary plane Euclidean and non-Euclidean geometries",
1352:
771:
767:
803:
531:
1063:
130:
363:{\displaystyle 0\rightarrow \operatorname {Tor} IW(F)\rightarrow W(F)\rightarrow W(F^{\mathrm {py} })}
709:
468:
740:
636:
579:
251:
156:
21:
1249:, Translated from the 2nd Japanese edition, paperback version of the 1977 edition (1st ed.),
220:
1219:
1172:
1096:
1425:
1395:
1366:
1330:
1291:
1254:
1211:
1164:
1120:
1080:
1058:
704:
623:
279:
186:
92:
25:
1435:
1405:
1301:
1238:
1227:
1203:
1191:
1156:
1130:
1088:
1072:
464:
1344:
1268:
1184:
646:
376:
1439:
1409:
1391:
1362:
1340:
1305:
1264:
1231:
1180:
1134:
1092:
791:
474:
206:
1280:
1243:
673:
408:
275:
257:
136:
112:
35:
1449:
1312:
1275:
1142:
1100:
240:
210:
1223:
1316:
1379:
846:
763:
247:
1215:
1207:
1168:
1084:
1250:
1145:(1972), "Quadratic forms over formally real fields and pythagorean fields",
1054:
1176:
1076:
1115:, Mathematical Surveys and Monographs, vol. 124, Providence, RI:
1160:
24:
in which every sum of two squares is a square: equivalently it has
1420:, London Mathematical Society Lecture Note Series, vol. 171,
1286:, CBMS Regional Conference Series in Mathematics, vol. 52,
635:
Pythagorean fields can be used to construct models for some of
507:
845:
is a formally real field in which the set of squares forms a
1059:"Die Legendre'schen Sätze über die Winkelsumme im Dreieck"
250:
of a
Pythagorean field is of order 2 if the field is not
849:. A superpythagorean field is necessarily Pythagorean.
1315:(2005), "Chapter VIII section 4: Pythagorean fields",
806:
159:
1245:
Encyclopedic dictionary of mathematics, Volumes I, II
743:
712:
676:
649:
534:. Please help to ensure that disputed statements are
477:
431:
411:
379:
290:
260:
223:
139:
115:
95:
58:
38:
852:The analogue of the Diller–Dress theorem holds: if
825:is a formally real field with the property that if
1279:
1242:
754:
729:
703:, such as the Pythagorean closure of the field of
682:
662:
492:
455:
417:
397:
362:
266:
231:
177:
145:
121:
101:
81:
44:
52:is an extension obtained by adjoining an element
1388:Ergebnisse der Mathematik und ihrer Grenzgebiete
989:
640:
405:is the fundamental ideal of the Witt ring of
133:taking Pythagorean extensions. For any field
8:
1318:Introduction to quadratic forms over fields
643:, 163 C). The coordinate geometry given by
254:, and torsion-free otherwise. For a field
243:Pythagorean field is quadratically closed.
880:and contained in the quadratic closure of
737:in one variable over the rational numbers
197:is the minimal ordered Pythagorean field.
1282:Orderings, valuations and quadratic forms
952:Milnor & Husemoller (1973) p. 72
943:Milnor & Husemoller (1973) p. 66
907:Milnor & Husemoller (1973) p. 71
744:
742:
713:
711:
675:
654:
648:
554:Learn how and when to remove this message
476:
456:{\displaystyle \operatorname {Tor} IW(F)}
430:
410:
378:
347:
346:
289:
259:
224:
222:
217:is Pythagorean field but not conversely (
165:
164:
158:
138:
114:
94:
71:
59:
57:
37:
1009:
1007:
530:Relevant discussion may be found on the
82:{\displaystyle {\sqrt {1+\lambda ^{2}}}}
897:
903:
901:
153:there is a minimal Pythagorean field
7:
876:is a formally real field containing
841:. An equivalent definition is that
566:The following conditions on a field
1359:Undergraduate Texts in Mathematics
1325:, vol. 67, Providence, R.I.:
1241:; Kawada, Yukiyosi, eds. (1980) ,
1109:Valuations, orderings, and Milnor
351:
348:
169:
166:
14:
868:. In the opposite direction, if
745:
714:
512:
239:is Pythagorean); however, a non
225:
129:. So a Pythagorean field is one
1323:Graduate Studies in Mathematics
1148:American Journal of Mathematics
864:is superpythagorean then so is
730:{\displaystyle \mathbf {Q} (x)}
833:and does not contain −1, then
724:
718:
487:
481:
450:
444:
392:
386:
357:
339:
333:
330:
324:
318:
315:
309:
294:
178:{\textstyle F^{\mathrm {py} }}
1:
1327:American Mathematical Society
1288:American Mathematical Society
1117:American Mathematical Society
755:{\displaystyle \mathbf {Q} ,}
701:non-archimedean ordered field
699:The Pythagorean closure of a
829:is a subgroup of index 2 in
810:, which is not Pythagorean.
782:This theorem states that if
232:{\displaystyle \mathbf {R} }
798:is Pythagorean, then so is
622:is the intersection of its
1472:
1422:Cambridge University Press
1351:Martin, George E. (1998),
860:is a finite extension and
604:then there is an order on
215:quadratically closed field
1382:; Husemoller, D. (1973),
990:Iyanaga & Kawada 1980
641:Iyanaga & Kawada 1980
1384:Symmetric Bilinear Forms
1208:10.4169/000298910x480063
925:Martin (1998) p. 89
872:is superpythagorean and
802:. As a consequence, no
102:{\displaystyle \lambda }
1416:Rajwade, A. R. (1993),
1354:Geometric Constructions
837:defines an ordering on
814:Superpythagorean fields
772:semi-Euclidean geometry
768:non-Legendrian geometry
820:superpythagorean field
804:algebraic number field
756:
731:
684:
664:
494:
457:
419:
399:
364:
268:
233:
185:containing it, unique
179:
147:
123:
103:
83:
46:
30:Pythagorean extension
1064:Mathematische Annalen
888:is superpythagorean.
757:
732:
685:
665:
663:{\displaystyle F^{n}}
616:have different signs.
504:Equivalent conditions
495:
458:
420:
400:
398:{\displaystyle IW(F)}
365:
269:
234:
180:
148:
124:
104:
84:
47:
1329:, pp. 255–264,
1192:Greenberg, Marvin J.
934:Rajwade (1993) p.230
778:Diller–Dress theorem
741:
710:
674:
647:
523:factual accuracy is
493:{\displaystyle W(F)}
475:
429:
409:
377:
288:
258:
221:
157:
137:
113:
93:
56:
36:
1456:Field (mathematics)
1107:Efrat, Ido (2006),
600:is not a square in
574:being Pythagorean:
467:(which is just the
191:Pythagorean closure
1077:10.1007/BF01448980
979:Efrat (2005) p.178
752:
727:
705:rational functions
680:
660:
631:Models of geometry
624:Euclidean closures
570:are equivalent to
490:
453:
415:
395:
360:
264:
229:
175:
143:
119:
99:
79:
42:
1336:978-0-8218-1095-8
1260:978-0-262-59010-5
1239:Iyanaga, Shôkichi
683:{\displaystyle F}
564:
563:
556:
418:{\displaystyle F}
267:{\displaystyle F}
187:up to isomorphism
146:{\displaystyle F}
122:{\displaystyle F}
77:
45:{\displaystyle F}
26:Pythagoras number
18:Pythagorean field
1463:
1442:
1412:
1390:, vol. 73,
1375:
1347:
1308:
1285:
1271:
1248:
1234:
1187:
1155:(4): 1155–1194,
1141:Elman, Richard;
1137:
1103:
1041:
1038:
1032:
1029:
1023:
1022:Lam (2005) p.269
1020:
1014:
1011:
1002:
999:
993:
986:
980:
977:
971:
970:Lam (2005) p.293
968:
962:
961:Lam (2005) p.410
959:
953:
950:
944:
941:
935:
932:
926:
923:
917:
916:Greenberg (2010)
914:
908:
905:
761:
759:
758:
753:
748:
736:
734:
733:
728:
717:
689:
687:
686:
681:
669:
667:
666:
661:
659:
658:
637:Hilbert's axioms
559:
552:
548:
545:
539:
536:reliably sourced
516:
515:
508:
499:
497:
496:
491:
465:torsion subgroup
462:
460:
459:
454:
424:
422:
421:
416:
404:
402:
401:
396:
369:
367:
366:
361:
356:
355:
354:
273:
271:
270:
265:
238:
236:
235:
230:
228:
184:
182:
181:
176:
174:
173:
172:
152:
150:
149:
144:
128:
126:
125:
120:
108:
106:
105:
100:
88:
86:
85:
80:
78:
76:
75:
60:
51:
49:
48:
43:
1471:
1470:
1466:
1465:
1464:
1462:
1461:
1460:
1446:
1445:
1432:
1415:
1402:
1392:Springer-Verlag
1378:
1373:
1363:Springer-Verlag
1350:
1337:
1311:
1298:
1274:
1261:
1237:
1190:
1161:10.2307/2373568
1140:
1127:
1106:
1053:
1050:
1045:
1044:
1040:Lam (1983) p.48
1039:
1035:
1031:Lam (1983) p.47
1030:
1026:
1021:
1017:
1013:Lam (1983) p.45
1012:
1005:
1000:
996:
987:
983:
978:
974:
969:
965:
960:
956:
951:
947:
942:
938:
933:
929:
924:
920:
915:
911:
906:
899:
894:
816:
792:field extension
780:
739:
738:
708:
707:
672:
671:
650:
645:
644:
633:
560:
549:
543:
540:
529:
521:This section's
517:
513:
506:
473:
472:
427:
426:
407:
406:
375:
374:
342:
286:
285:
256:
255:
219:
218:
207:Euclidean field
203:
160:
155:
154:
135:
134:
111:
110:
91:
90:
67:
54:
53:
34:
33:
28:equal to 1. A
12:
11:
5:
1469:
1467:
1459:
1458:
1448:
1447:
1444:
1443:
1430:
1413:
1400:
1376:
1371:
1348:
1335:
1309:
1296:
1272:
1259:
1235:
1202:(3): 198–219,
1196:Am. Math. Mon.
1188:
1138:
1125:
1104:
1071:(3): 404–439,
1049:
1046:
1043:
1042:
1033:
1024:
1015:
1003:
994:
981:
972:
963:
954:
945:
936:
927:
918:
909:
896:
895:
893:
890:
815:
812:
779:
776:
766:, examples of
751:
747:
726:
723:
720:
716:
679:
657:
653:
639:for geometry (
632:
629:
628:
627:
617:
594:
562:
561:
520:
518:
511:
505:
502:
489:
486:
483:
480:
452:
449:
446:
443:
440:
437:
434:
414:
394:
391:
388:
385:
382:
371:
370:
359:
353:
350:
345:
341:
338:
335:
332:
329:
326:
323:
320:
317:
314:
311:
308:
305:
302:
299:
296:
293:
278:involving the
276:exact sequence
263:
227:
202:
199:
171:
168:
163:
142:
118:
98:
74:
70:
66:
63:
41:
16:In algebra, a
13:
10:
9:
6:
4:
3:
2:
1468:
1457:
1454:
1453:
1451:
1441:
1437:
1433:
1431:0-521-42668-5
1427:
1423:
1419:
1414:
1411:
1407:
1403:
1401:3-540-06009-X
1397:
1393:
1389:
1385:
1381:
1377:
1374:
1372:0-387-98276-0
1368:
1364:
1360:
1356:
1355:
1349:
1346:
1342:
1338:
1332:
1328:
1324:
1320:
1319:
1314:
1310:
1307:
1303:
1299:
1297:0-8218-0702-1
1293:
1289:
1284:
1283:
1277:
1273:
1270:
1266:
1262:
1256:
1252:
1247:
1246:
1240:
1236:
1233:
1229:
1225:
1221:
1217:
1213:
1209:
1205:
1201:
1197:
1193:
1189:
1186:
1182:
1178:
1174:
1170:
1166:
1162:
1158:
1154:
1150:
1149:
1144:
1139:
1136:
1132:
1128:
1126:0-8218-4041-X
1122:
1118:
1114:
1110:
1105:
1102:
1098:
1094:
1090:
1086:
1082:
1078:
1074:
1070:
1066:
1065:
1060:
1056:
1052:
1051:
1047:
1037:
1034:
1028:
1025:
1019:
1016:
1010:
1008:
1004:
998:
995:
991:
985:
982:
976:
973:
967:
964:
958:
955:
949:
946:
940:
937:
931:
928:
922:
919:
913:
910:
904:
902:
898:
891:
889:
887:
883:
879:
875:
871:
867:
863:
859:
855:
850:
848:
844:
840:
836:
832:
828:
824:
821:
813:
811:
809:
805:
801:
797:
793:
789:
785:
777:
775:
773:
769:
765:
749:
721:
706:
702:
697:
694:
677:
655:
651:
642:
638:
630:
625:
621:
618:
615:
611:
607:
603:
599:
595:
592:
588:
585:
583:
577:
576:
575:
573:
569:
558:
555:
547:
537:
533:
527:
526:
519:
510:
509:
503:
501:
484:
478:
470:
466:
447:
441:
438:
435:
432:
412:
389:
383:
380:
343:
336:
327:
321:
312:
306:
303:
300:
297:
291:
284:
283:
282:
281:
277:
261:
253:
252:formally real
249:
244:
242:
241:formally real
216:
212:
211:ordered field
208:
200:
198:
196:
195:Hilbert field
192:
189:, called its
188:
161:
140:
132:
116:
96:
72:
68:
64:
61:
39:
31:
27:
23:
19:
1417:
1383:
1353:
1317:
1281:
1244:
1199:
1195:
1152:
1146:
1112:
1108:
1068:
1062:
1036:
1027:
1018:
997:
984:
975:
966:
957:
948:
939:
930:
921:
912:
885:
881:
877:
873:
869:
865:
861:
857:
853:
851:
842:
838:
834:
830:
826:
822:
819:
817:
807:
799:
795:
790:is a finite
787:
783:
781:
698:
692:
634:
619:
613:
609:
605:
601:
597:
593:) is 0 or 1.
590:
586:
581:
571:
567:
565:
550:
541:
522:
463:denotes its
372:
274:there is an
245:
204:
194:
190:
131:closed under
29:
17:
15:
1001:Dehn (1900)
764:Dehn planes
32:of a field
1440:0785.11022
1410:0292.10016
1380:Milnor, J.
1313:Lam, T. Y.
1306:0516.12001
1276:Lam, T. Y.
1232:1206.51015
1143:Lam, T. Y.
1135:1103.12002
1093:31.0471.01
1048:References
608:for which
584:-invariant
469:nilradical
280:Witt rings
201:Properties
1251:MIT Press
1216:0002-9890
1169:0002-9327
1101:122651688
1085:0025-5831
1055:Dehn, Max
544:June 2023
532:talk page
436:
334:→
319:→
301:
295:→
248:Witt ring
97:λ
89:for some
69:λ
1450:Category
1278:(1983),
1057:(1900),
992:, 163 D)
580:general
525:disputed
1418:Squares
1345:2104929
1269:0591028
1224:7792750
1185:0314878
1177:2373568
1113:-theory
193:. The
1438:
1428:
1408:
1398:
1369:
1343:
1333:
1304:
1294:
1267:
1257:
1230:
1222:
1214:
1183:
1175:
1167:
1133:
1123:
1099:
1091:
1083:
794:, and
373:where
205:Every
1220:S2CID
1173:JSTOR
1097:S2CID
892:Notes
884:then
22:field
20:is a
1426:ISBN
1396:ISBN
1367:ISBN
1331:ISBN
1292:ISBN
1255:ISBN
1212:ISSN
1165:ISSN
1121:ISBN
1081:ISSN
770:and
670:for
578:The
425:and
246:The
209:(an
1436:Zbl
1406:Zbl
1302:Zbl
1228:Zbl
1204:doi
1200:117
1157:doi
1131:Zbl
1089:JFM
1073:doi
847:fan
596:If
500:).
471:of
433:Tor
298:Tor
109:in
1452::
1434:,
1424:,
1404:,
1394:,
1386:,
1365:,
1361:,
1357:,
1341:MR
1339:,
1321:,
1300:,
1290:,
1265:MR
1263:,
1253:,
1226:,
1218:,
1210:,
1198:,
1181:MR
1179:,
1171:,
1163:,
1153:94
1151:,
1129:,
1119:,
1095:,
1087:,
1079:,
1069:53
1067:,
1061:,
1006:^
900:^
818:A
612:,
598:ab
1206::
1159::
1111:K
1075::
988:(
886:E
882:F
878:F
874:E
870:F
866:F
862:E
858:F
856:/
854:E
843:F
839:F
835:S
831:F
827:S
823:F
808:Q
800:F
796:E
788:F
786:/
784:E
750:,
746:Q
725:)
722:x
719:(
715:Q
693:F
678:F
656:n
652:F
626:.
620:F
614:b
610:a
606:F
602:F
591:F
589:(
587:u
582:u
572:F
568:F
557:)
551:(
546:)
542:(
538:.
528:.
488:)
485:F
482:(
479:W
451:)
448:F
445:(
442:W
439:I
413:F
393:)
390:F
387:(
384:W
381:I
358:)
352:y
349:p
344:F
340:(
337:W
331:)
328:F
325:(
322:W
316:)
313:F
310:(
307:W
304:I
292:0
262:F
226:R
170:y
167:p
162:F
141:F
117:F
73:2
65:+
62:1
40:F
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.