312:
by rotations in the plane is irreducible (over the field of real numbers), but is not absolutely irreducible. After extending the field to complex numbers, it splits into two irreducible components. This is to be expected, since the circle group is
591:
Therefore, this algebraic variety consists of two lines intersecting at the origin and is not absolutely irreducible. This holds either already over the ground field, if −1 is a square, or over the quadratic extension obtained by adjoining
562:
221:
468:
370:
95:
135:
582:
739:
709:
682:
652:
625:
317:
and it is known that all irreducible representations of commutative groups over an algebraically closed field are one-dimensional.
766:
285:
243:
761:
410:
254:, which emphasizes that the coefficients of the defining equations may not belong to an algebraically closed field.
485:
144:
32:
387:
44:
284:
A univariate polynomial of degree greater than or equal to 2 is never absolutely irreducible, due to the
261:
386:
over the field of complex numbers. Absolute irreducibility more generally holds over any field not of
235:
725:
427:
329:
54:
242:
is absolutely irreducible if it is not the union of two algebraic sets defined by equations in an
672:
100:
735:
705:
678:
648:
621:
272:
251:
729:
699:
642:
615:
302:
292:
265:
36:
567:
138:
17:
755:
383:
271:
In all cases, being absolutely irreducible is the same as being irreducible over the
48:
407:
309:
230:
is absolutely irreducible if it is irreducible over every algebraic extension of
314:
28:
250:. In other words, an absolutely irreducible algebraic set is a synonym of an
476:
is not absolutely irreducible. Indeed, the left hand side can be factored as
620:, Pure and Applied Mathematics, vol. 20, Academic Press, p. 10,
137:
is irreducible over the integers and the reals, it is reducible over the
379:
641:
Grabmeier, Johannes; Kaltofen, Erich; Weispfenning, Volker (2003),
704:, Monographs in Contemporary Mathematics, Springer, p. 53,
644:
Computer
Algebra Handbook: Foundations, Applications, Systems
390:
two. In characteristic two, the equation is equivalent to (
291:
The irreducible two-dimensional representation of the
570:
488:
430:
332:
147:
103:
57:
731:
Algebraic
Geometry in Coding Theory and Cryptography
576:
556:
462:
364:
320:The real algebraic variety defined by the equation
238:defined by equations with coefficients in a field
226:More generally, a polynomial defined over a field
215:
129:
89:
677:(2nd ed.), CRC Press, pp. 8–17 – 8-18,
301:of order 6, originally defined over the field of
398: −1) = 0. Hence it defines the double line
378:is absolutely irreducible. It is the ordinary
8:
614:Borevich, Z. I.; Shafarevich, I. R. (1986),
418:The algebraic variety given by the equation
260:is also applied, with the same meaning, to
734:, Princeton University Press, p. 47,
382:over the reals and remains an irreducible
569:
557:{\displaystyle x^{2}+y^{2}=(x+yi)(x-yi),}
506:
493:
487:
448:
435:
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216:{\displaystyle x^{2}+y^{2}=(x+iy)(x-iy),}
165:
152:
146:
121:
108:
102:
75:
62:
56:
666:
664:
606:
223:and thus not absolutely irreducible.
97:is absolutely irreducible, but while
7:
25:
701:Arithmetic of Algebraic Curves
548:
533:
530:
515:
286:fundamental theorem of algebra
244:algebraically closed extension
207:
192:
189:
174:
1:
698:Stepanov, Serguei A. (1994),
463:{\displaystyle x^{2}+y^{2}=0}
365:{\displaystyle x^{2}+y^{2}=1}
90:{\displaystyle x^{2}+y^{2}-1}
305:, is absolutely irreducible.
130:{\displaystyle x^{2}+y^{2}}
783:
308:The representation of the
728:; Xing, Chaoping (2009),
674:Computer Science Handbook
671:Tucker, Allen B. (2004),
647:, Springer, p. 26,
584:is a square root of −1.
33:multivariate polynomial
578:
558:
464:
366:
262:linear representations
258:Absolutely irreducible
217:
131:
91:
41:absolutely irreducible
18:Absolutely irreducible
767:Representation theory
579:
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465:
406: =1, which is a
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275:of the ground field.
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132:
92:
726:Niederreiter, Harald
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236:affine algebraic set
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101:
55:
762:Algebraic geometry
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577:{\displaystyle i}
273:algebraic closure
252:algebraic variety
35:defined over the
16:(Redirected from
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303:rational numbers
266:algebraic groups
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37:rational numbers
21:
782:
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293:symmetric group
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139:complex numbers
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51:. For example,
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12:
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5:
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388:characteristic
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24:
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3:
2:
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741:9781400831302
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711:9780306110368
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684:9780203494455
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654:9783540654667
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629:
627:9780080873329
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619:
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617:Number theory
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385:
384:conic section
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171:
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149:
140:
122:
118:
114:
109:
105:
84:
81:
76:
72:
68:
63:
59:
50:
49:complex field
46:
42:
38:
34:
30:
19:
730:
720:
700:
693:
673:
643:
636:
616:
609:
593:
403:
399:
395:
391:
310:circle group
295:
270:
257:
256:
247:
239:
231:
227:
225:
40:
26:
408:non-reduced
315:commutative
45:irreducible
29:mathematics
756:Categories
601:References
43:if it is
540:−
234:, and an
199:−
82:−
47:over the
279:Examples
738:
708:
681:
651:
624:
564:where
411:scheme
380:circle
736:ISBN
706:ISBN
679:ISBN
649:ISBN
622:ISBN
31:, a
264:of
246:of
141:as
39:is
27:In
758::
663:^
268:.
745:.
715:.
688:.
658:.
631:.
596:.
594:i
572:i
552:,
549:)
546:i
543:y
537:x
534:(
531:)
528:i
525:y
522:+
519:x
516:(
513:=
508:2
504:y
500:+
495:2
491:x
458:0
455:=
450:2
446:y
442:+
437:2
433:x
413:.
404:y
400:x
396:y
392:x
360:1
357:=
352:2
348:y
344:+
339:2
335:x
299:3
296:S
288:.
248:K
240:K
232:K
228:K
211:,
208:)
205:y
202:i
196:x
193:(
190:)
187:y
184:i
181:+
178:x
175:(
172:=
167:2
163:y
159:+
154:2
150:x
123:2
119:y
115:+
110:2
106:x
85:1
77:2
73:y
69:+
64:2
60:x
20:)
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