251:
As a consequence, the
Pythagoras number of a non-formally-real field is either a power of 2, or 1 more than a power of 2. All such cases occur: i.e., for each pair (
576:
537:
133:
529:
17:
603:
568:
288:
376:
122:
31:
547:
586:
555:
8:
159:
54:
93:
is a field with
Pythagoras number 1: that is, every sum of squares is already a square.
572:
533:
393:
89:
35:
608:
582:
551:
57:
describes the structure of the set of squares in the field. The
Pythagoras number
543:
166:
137:
597:
332:
118:
27:
Number which describes the structure of the set of squares in a given field
103:
42:
125:, not every element is a square, but all are the sum of two squares, so
140:
is a sum of four squares, and not all are sums of three squares, so
567:. London Mathematical Society Lecture Note Series. Vol. 171.
70:
158:
Every positive integer occurs as the
Pythagoras number of some
259:) of the form (2,2) or (2,2 + 1), there exists a field
411:) is the smallest power of 2 which is not less than
213:) + 1, and both cases are possible: for
595:
532:. Vol. 67. American Mathematical Society.
526:Introduction to Quadratic Forms over Fields
392:The Pythagoras number is related to the
295:) and fields of characteristic 2 (e.g.,
165:The Pythagoras number is related to the
562:
14:
596:
504:
495:
459:
450:
523:
486:
477:
24:
468:
76:such that every sum of squares in
25:
620:
530:Graduate Studies in Mathematics
134:Lagrange's four-square theorem
13:
1:
517:
152:
7:
289:quadratically closed fields
96:
10:
625:
569:Cambridge University Press
492:Rajwade (1993) p. 261
483:Rajwade (1993) p. 228
423:is not formally real then
189:is not formally real then
29:
474:Rajwade (1993) p. 44
69:is the smallest positive
443:
30:Not to be confused with
563:Rajwade, A. R. (1993).
524:Lam, Tsit-Yuen (2005).
346:give (1,2); for primes
318:)) = (1,1); for primes
510:Lam (2005) p. 395
501:Lam (2005) p. 396
465:Lam (2005) p. 398
403:is formally real then
456:Lam (2005) p. 36
375:gives (4,4), and the
185:) + 1. If
32:Pythagoras's constant
604:Field (mathematics)
160:formally real field
102:Every non-negative
394:height of a field
359:gives (2,2), and
229:= 1, whereas for
136:, every positive
90:Pythagorean field
47:Pythagoras number
36:height of a field
16:(Redirected from
616:
590:
559:
511:
508:
502:
499:
493:
490:
484:
481:
475:
472:
466:
463:
457:
454:
287:). For example,
244: = 1,
148:) = 4.
114:) = 1.
106:is a square, so
21:
624:
623:
619:
618:
617:
615:
614:
613:
594:
593:
579:
540:
520:
515:
514:
509:
505:
500:
496:
491:
487:
482:
478:
473:
469:
464:
460:
455:
451:
446:
384:
374:
367:
358:
345:
330:
301:
248: = 2.
239:
155:
138:rational number
129: = 2.
99:
39:
28:
23:
22:
15:
12:
11:
5:
622:
612:
611:
606:
592:
591:
577:
560:
538:
519:
516:
513:
512:
503:
494:
485:
476:
467:
458:
448:
447:
445:
442:
441:
440:
390:
389:) gives (4,5).
382:
377:function field
372:
363:
354:
341:
326:
299:
249:
237:
163:
154:
151:
150:
149:
130:
123:characteristic
115:
98:
95:
51:reduced height
26:
18:Reduced height
9:
6:
4:
3:
2:
621:
610:
607:
605:
602:
601:
599:
588:
584:
580:
578:0-521-42668-5
574:
570:
566:
561:
557:
553:
549:
545:
541:
539:0-8218-1095-2
535:
531:
527:
522:
521:
507:
498:
489:
480:
471:
462:
453:
449:
438:
434:
430:
426:
422:
418:
414:
410:
406:
402:
398:
395:
391:
388:
381:
378:
371:
368:gives (2,3);
366:
362:
357:
353:
350:≡ 3 (mod 4),
349:
344:
340:
337:
335:
329:
325:
322:≡ 1 (mod 4),
321:
317:
313:
309:
305:
298:
294:
290:
286:
282:
278:
274:
270:
266:
262:
258:
254:
250:
247:
243:
236:
232:
228:
224:
220:
216:
212:
208:
204:
200:
196:
192:
188:
184:
180:
176:
172:
168:
164:
161:
157:
156:
147:
143:
139:
135:
131:
128:
124:
120:
116:
113:
109:
105:
101:
100:
94:
92:
91:
85:
83:
79:
75:
72:
68:
65:) of a field
64:
60:
56:
52:
48:
44:
37:
33:
19:
564:
525:
506:
497:
488:
479:
470:
461:
452:
436:
432:
428:
424:
420:
416:
412:
408:
404:
400:
396:
386:
379:
369:
364:
360:
355:
351:
347:
342:
338:
333:
327:
323:
319:
315:
311:
307:
303:
296:
292:
284:
280:
276:
272:
268:
264:
260:
256:
252:
245:
241:
234:
230:
226:
222:
218:
214:
210:
206:
202:
198:
194:
190:
186:
182:
178:
174:
170:
145:
141:
126:
119:finite field
111:
107:
88:
86:
81:
80:is a sum of
77:
73:
66:
62:
58:
50:
46:
40:
336:-adic field
263:such that (
104:real number
43:mathematics
598:Categories
587:0785.11022
556:1068.11023
518:References
153:Properties
84:squares.
331:and the
302:) give (
240:we have
221:we have
97:Examples
609:Sumsets
565:Squares
548:2104929
291:(e.g.,
121:of odd
71:integer
585:
575:
554:
546:
536:
419:); if
279:)) = (
117:For a
45:, the
444:Notes
431:) = 2
399:: if
205:) ≤
177:) ≤
167:Stufe
55:field
53:of a
573:ISBN
534:ISBN
197:) ≤
169:by
583:Zbl
552:Zbl
132:By
49:or
41:In
34:or
600::
581:.
571:.
550:.
544:MR
542:.
528:.
439:).
310:),
271:),
233:=
225:=
217:=
87:A
589:.
558:.
437:F
435:(
433:s
429:F
427:(
425:h
421:F
417:F
415:(
413:p
409:F
407:(
405:h
401:F
397:F
387:X
385:(
383:2
380:Q
373:2
370:Q
365:p
361:Q
356:p
352:F
348:p
343:p
339:Q
334:p
328:p
324:F
320:p
316:F
314:(
312:p
308:F
306:(
304:s
300:2
297:F
293:C
285:p
283:,
281:s
277:F
275:(
273:p
269:F
267:(
265:s
261:F
257:p
255:,
253:s
246:p
242:s
238:5
235:F
231:F
227:p
223:s
219:C
215:F
211:F
209:(
207:s
203:F
201:(
199:p
195:F
193:(
191:s
187:F
183:F
181:(
179:s
175:F
173:(
171:p
162:.
146:Q
144:(
142:p
127:p
112:R
110:(
108:p
82:p
78:K
74:p
67:K
63:K
61:(
59:p
38:.
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.