216:
146:
314:
242:
347:
169:
580:"Zyklische Körper und Algebren der Characteristik p vom Grad p. Struktur diskret bewerteter perfekter Körper mit vollkommenem Restklassenkörper der Charakteristik p"
584:
364:
is the splitting field of an Artin–Schreier polynomial. This can be proved using additive counterparts of the methods involved in
549:
507:
182:
499:
411:
245:
108:
317:
17:
410:
of abelian varieties must, for their function fields, give either an Artin–Schreier extension or a
287:
36:
422:
There is an analogue of Artin–Schreier theory which describes cyclic extensions in characteristic
384:
369:
221:
613:
323:
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154:
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8:
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44:
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391:, representing one of the possible classes of extensions in a solvable chain.
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365:
56:
40:
32:
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435:
24:
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439:
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68:
52:
502:, vol. 211 (Revised third ed.), New York: Springer-Verlag,
465:
63:) introduced Artin–Schreier theory for extensions of prime degree
399:
463:(1927), "Eine Kennzeichnung der reell abgeschlossenen Körper",
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383:
Artin–Schreier extensions play a role in the theory of
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185:
157:
111:
75:) generalized it to extensions of prime power degree
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308:
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140:
417:
605:
585:Journal für die reine und angewandte Mathematik
542:Grundlehren der Mathematischen Wissenschaften
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211:{\displaystyle \alpha \neq \beta ^{p}-\beta }
60:
16:For the result about real-closed fields, see
536:; Schmidt, Alexander; Wingberg, Kay (2000),
544:, vol. 323, Berlin: Springer-Verlag,
471:, Springer Berlin / Heidelberg: 225–231,
137:
394:They also play a part in the theory of
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47:of degree equal to the characteristic
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443:
352:Conversely, any Galois extension of
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13:
272:. This follows since for any root
141:{\displaystyle X^{p}-X-\alpha ,\,}
14:
625:
430:-power degree (not just degree
376:. These extensions are called
360:equal to the characteristic of
418:Artin–Schreier–Witt extensions
336:
330:
1:
500:Graduate Texts in Mathematics
449:
309:{\displaystyle 1\leq i\leq p}
412:purely inseparable extension
7:
538:Cohomology of Number Fields
320:—so the splitting field is
237:{\displaystyle \beta \in K}
10:
630:
35:, specifically a positive
15:
598:10.1515/crll.1937.176.126
378:Artin–Schreier extensions
342:{\displaystyle K(\beta )}
177:Artin–Schreier polynomial
316:, form all the roots—by
406:, an isogeny of degree
385:solvability by radicals
318:Fermat's little theorem
164:{\displaystyle \alpha }
343:
310:
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212:
165:
142:
18:Artin–Schreier theorem
344:
311:
244:, this polynomial is
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213:
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143:
29:Artin–Schreier theory
402:. In characteristic
387:, in characteristic
370:Hilbert's theorem 90
324:
288:
222:
183:
155:
109:
477:10.1007/BF02952522
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306:
234:
208:
161:
138:
90:of characteristic
551:978-3-540-66671-4
509:978-0-387-95385-4
396:abelian varieties
374:Galois cohomology
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600:
570:
534:Neukirch, Jürgen
528:
487:
438:, developed by
434:itself), using
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346:
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315:
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262:cyclic extension
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321:
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254:splitting field
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184:
181:
180:
175:, is called an
156:
153:
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116:
112:
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31:is a branch of
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12:
11:
5:
627:
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572:
550:
530:
508:
488:
461:Schreier, Otto
451:
448:
419:
416:
338:
335:
332:
329:
305:
302:
299:
296:
293:
276:, the numbers
233:
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160:
149:
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136:
133:
130:
127:
124:
119:
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37:characteristic
9:
6:
4:
3:
2:
626:
615:
614:Galois theory
612:
611:
609:
599:
595:
591:
588:(in German),
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429:
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415:
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409:
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397:
392:
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386:
381:
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375:
372:and additive
371:
367:
366:Kummer theory
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359:
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319:
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297:
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134:
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125:
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105:
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97:
93:
89:
85:
80:
78:
74:
70:
66:
62:
58:
55: and
54:
50:
46:
43:, for Galois
42:
41:Kummer theory
38:
34:
33:Galois theory
30:
26:
19:
589:
583:
571:Section VI.1
541:
537:
529:Section VI.6
495:
468:
464:
436:Witt vectors
431:
427:
423:
421:
407:
403:
393:
388:
382:
377:
361:
357:
353:
351:
281:
277:
273:
269:
265:
257:
249:
176:
172:
150:
102:of the form
96:prime number
91:
83:
81:
76:
64:
48:
39:analogue of
28:
22:
592:: 126–140,
576:Witt, Ernst
492:Lang, Serge
457:Artin, Emil
246:irreducible
25:mathematics
568:0948.11001
526:0984.00001
450:References
398:and their
368:, such as
356:of degree
268:of degree
252:, and its
100:polynomial
45:extensions
485:0025-5858
400:isogenies
334:β
301:≤
295:≤
229:∈
226:β
206:β
203:−
194:β
190:≠
187:α
159:α
132:α
129:−
123:−
608:Category
578:(1936),
494:(2002),
218:for all
57:Schreier
560:1737196
518:1878556
496:Algebra
442: (
179:. When
71: (
59: (
566:
558:
548:
524:
516:
506:
483:
284:, for
98:, any
67:, and
260:is a
256:over
88:field
86:is a
53:Artin
546:ISBN
504:ISBN
481:ISSN
444:1936
440:Witt
151:for
94:, a
73:1936
69:Witt
61:1927
594:doi
590:176
564:Zbl
522:Zbl
473:doi
446:).
426:of
380:.
264:of
248:in
171:in
82:If
51:.
23:In
610::
582:,
562:,
556:MR
554:,
540:,
520:,
514:MR
512:,
498:,
479:,
467:,
459:;
414:.
349:.
280:+
79:.
27:,
596::
475::
469:5
432:p
428:p
424:p
408:p
404:p
389:p
362:K
358:p
354:K
337:)
331:(
328:K
304:p
298:i
292:1
282:i
278:β
274:β
270:p
266:K
258:K
250:K
232:K
198:p
173:K
135:,
126:X
118:p
114:X
92:p
84:K
77:p
65:p
49:p
20:.
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