2118:
2431:
1856:
2189:
2113:{\displaystyle {\begin{aligned}\mathbb {Q} \left({\sqrt {2}},{\sqrt {3}}\right)&=\mathbb {Q} \left({\sqrt {2}}\right)\left({\sqrt {3}}\right)\\&=\left\{a+b{\sqrt {3}}\mid a,b\in \mathbb {Q} \left({\sqrt {2}}\right)\right\}\\&=\left\{a+b{\sqrt {2}}+c{\sqrt {3}}+d{\sqrt {6}}\mid a,b,c,d\in \mathbb {Q} \right\},\end{aligned}}}
2426:{\displaystyle {\begin{aligned}\mathbb {Q} ({\sqrt {2}},{\sqrt {3}})&=\mathbb {Q} ({\sqrt {2}}+{\sqrt {3}})\\&=\left\{a+b({\sqrt {2}}+{\sqrt {3}})+c({\sqrt {2}}+{\sqrt {3}})^{2}+d({\sqrt {2}}+{\sqrt {3}})^{3}\mid a,b,c,d\in \mathbb {Q} \right\}.\end{aligned}}}
5145:. Extension of scalars of polynomials is often used implicitly, by just considering the coefficients as being elements of a larger field, but may also be considered more formally. Extension of scalars has numerous applications, as discussed in
1777:
1331:
It is often desirable to talk about field extensions in situations where the small field is not actually contained in the larger one, but is naturally embedded. For this purpose, one abstractly defines a field extension as an
2842:
5098:
is exactly the field. For example, the only finite field extension of the real numbers is the complex numbers, while the quaternions are a central simple algebra over the reals, and all CSAs over the reals are
4287:
4028:). On the opposite, even when one knows a transcendence basis, it may be difficult to decide whether the extension is purely separable, and if it is so, it may be difficult to find a transcendence basis
3539:
2194:
1861:
4560:
4478:
4095:
2767:
1635:
1845:
1164:
3291:
2156:
1593:
1487:
1045:
1100:
4962:
5323:
2961:
2496:
1680:
4210:
4140:
3634:
2653:
2181:
1805:
4422:
3777:
3751:
3729:
3707:
3685:
3561:
2983:
2456:
1455:
1433:
1411:
1386:
245:
88:
3609:
3585:
3111:
4692:
3144:
862:
4656:
2575:
1695:
1561:
4623:
3642:
it is finite. This implies that an extension is algebraic if and only if it is the union of its finite subextensions, and that every finite extension is algebraic.
4586:
4504:
4400:
5040:
4920:
4858:
4827:
4763:
3959:
3919:
3819:
3486:
3071:
893:
788:
760:
732:
704:
676:
567:
536:
469:
59:
4374:
1529:
4712:
4351:
4327:
4307:
4160:
4115:
398:
360:
340:
313:
285:
265:
216:
193:
622:
2779:
5054:). The significance of Galois extensions and Galois groups is that they allow a complete description of the intermediate fields: there is a
4588:
is a transcendence basis that does not generates the whole extension. However the extension is purely transcendental since, if one set
2466:. Another extension field of the rationals, which is also important in number theory, although not a finite extension, is the field of
3985:, but, however, there are extensions with this property which are not purely transcendental—a class of such extensions take the form
4215:
3211:
is a finite extension. In this case the degree of the extension equals the degree of the minimal polynomial, and a basis of the
4718:
3005:
5356:
5063:
3499:
4729:
of a rational variety is equivalent with the problem of finding a transcendence basis that generates the whole extension.
5331:
4509:
4427:
5129:" on associated algebraic objects. For example, given a real vector space, one can produce a complex vector space via
3893:. All transcendence bases have the same cardinality, equal to the transcendence degree of the extension. An extension
5309:
4050:
3167:
2711:
1598:
1215:
1810:
1105:
3229:
2126:
1566:
1460:
992:
1358:
Henceforth, we will suppress the injective homomorphism and assume that we are dealing with actual subfields.
5402:
5348:
3496:. Equivalently, an algebraic extension is an extension that is generated by algebraic elements. For example,
373:, which is itself a subfield of the complex numbers. More generally, the field of rational numbers is (or is
1320:
or any other kind of division. Instead the slash expresses the word "over". In some literature the notation
1050:
1683:
5392:
627:
The degree of an extension is 1 if and only if the two fields are equal. In this case, the extension is a
5397:
586:
5146:
5158:
4928:
1343:
non-zero ring homomorphism between fields is injective because fields do not possess nontrivial proper
2932:
2472:
1639:
5419:
4888:
3830:
1256:
98:
4169:
5163:
4726:
4120:
3788:
3614:
2596:
2161:
1785:
1252:
582:
404:
378:
4405:
3760:
3734:
3712:
3690:
3668:
3544:
2966:
2439:
1438:
1416:
1394:
1369:
224:
67:
3590:
3566:
3092:
149:
5351:, vol. 211 (Corrected fourth printing, revised third ed.), New York: Springer-Verlag,
1772:{\displaystyle \mathbb {Q} ({\sqrt {2}})=\left\{a+b{\sqrt {2}}\mid a,b\in \mathbb {Q} \right\},}
5087:
4770:
3175:
2578:
2459:
292:
4661:
3116:
796:
4628:
2547:
1534:
288:
4591:
5142:
5126:
5120:
5095:
4873:
4565:
4483:
4379:
3846:
2922:
8:
5134:
5017:
4897:
4861:
4835:
4804:
4740:
3936:
3896:
3882:
3796:
3463:
3048:
3032:
2586:
1344:
870:
765:
737:
709:
681:
653:
544:
513:
446:
196:
62:
36:
4356:
3661:, and also the smallest extension field such that every polynomial with coefficients in
1492:
5079:
4891:
states that every finite separable extension has a primitive element (i.e. is simple).
4722:
4697:
4336:
4312:
4292:
4145:
4100:
3078:
2900:
2664:
2521:
1352:
1333:
383:
345:
325:
298:
270:
250:
201:
178:
161:
595:
5374:
5352:
5327:
5305:
5178:
5173:
5100:
5008:
4330:
3309:
3036:
2994:
2986:
2864:
2183:
of degree 2 and 4 respectively. It is also a simple extension, as one can show that
1336:
1285:
5130:
4881:
4766:
3159:
1207:
153:
2837:{\displaystyle \operatorname {GF} (p)=\mathbb {F} _{p}=\mathbb {Z} /p\mathbb {Z} }
5370:
5168:
5104:
5075:
4163:
2915:
2675:
2514:
366:
18:
Construction of a larger algebraic field by "adding elements" to a smaller field
5301:
5091:
3639:
1317:
974:
138:
134:
4016:
is a transcendence basis of the extension, it doesn't necessarily follow that
291:
under the operations of addition, subtraction, multiplication, and taking the
5413:
5004:
2590:
2510:
2467:
2463:
1313:
157:
3845:. The largest cardinality of an algebraically independent set is called the
5000:
4965:
4717:
Purely transcendental extensions of an algebraically closed field occur as
3216:
2706:
2698:
578:
5133:. In addition to vector spaces, one can perform extension of scalars for
2774:
1389:
374:
370:
142:
22:
5340:
5138:
5108:
3082:
2884:
2690:
in which the given polynomial splits into a product of linear factors.
5055:
4999:. When the extension is Galois this automorphism group is called the
407:
of a subfield is the same as the characteristic of the larger field.
5103:
to the reals or the quaternions. CSAs can be further generalized to
247:
that is a field with respect to the field operations inherited from
5094:(no non-trivial 2-sided ideals, just as for a field) and where the
5059:
1348:
130:
1259:, which does not hold true for fields of non-zero characteristic.
5083:
26:
5378:
3113:
is algebraic over the rational numbers, because it is a root of
4506:
is a transcendence basis that does not generates the extension
1435:
in turn is an extension field of the field of rational numbers
219:
2505:
It is common to construct an extension field of a given field
3650:
2705:
is a positive integer, there is a unique (up to isomorphism)
1255:
0, every finite extension is a simple extension. This is the
4282:{\displaystyle \mathbb {Q} (X)/\langle Y^{2}-X^{3}\rangle ,}
2907:. This field of rational functions is an extension field of
4876:, i.e., has no repeated roots in an algebraic closure over
3973:). Such an extension has the property that all elements of
3837:
if no non-trivial polynomial relation with coefficients in
2985:
if we identify every complex number with the corresponding
4732:
145:; the real numbers are a subfield of the complex numbers.
2674:
By iterating the above construction, one can construct a
1807:
also clearly a simple extension. The degree is 2 because
4884:
is a field extension that is both normal and separable.
3320:. This results from the preceding characterization: if
1312:
is purely formal and does not imply the formation of a
5042:, one is often interested in the intermediate fields
5020:
4931:
4900:
4838:
4807:
4743:
4700:
4664:
4631:
4594:
4568:
4512:
4486:
4430:
4408:
4382:
4359:
4339:
4315:
4295:
4218:
4172:
4148:
4123:
4103:
4053:
3939:
3899:
3799:
3763:
3737:
3715:
3693:
3671:
3617:
3593:
3569:
3547:
3534:{\displaystyle \mathbb {Q} ({\sqrt {2}},{\sqrt {3}})}
3502:
3466:
3232:
3119:
3095:
3051:
2969:
2935:
2782:
2714:
2599:
2550:
2475:
2442:
2192:
2164:
2129:
1859:
1813:
1788:
1698:
1642:
1601:
1569:
1537:
1495:
1463:
1441:
1419:
1397:
1372:
1108:
1053:
995:
873:
799:
768:
740:
712:
684:
656:
598:
547:
516:
449:
386:
348:
328:
301:
273:
267:. Equivalently, a subfield is a subset that contains
253:
227:
204:
181:
70:
39:
5107:, where the base field is replaced by a commutative
5003:of the extension. Extensions whose Galois group is
4829:is normal and which is minimal with this property.
3015:), consisting of the rational functions defined on
5034:
4956:
4914:
4852:
4821:
4757:
4706:
4686:
4650:
4617:
4580:
4554:
4498:
4472:
4416:
4394:
4368:
4345:
4321:
4301:
4281:
4204:
4154:
4134:
4109:
4089:
3953:
3929:if and only if there exists a transcendence basis
3913:
3813:
3771:
3745:
3723:
3701:
3679:
3628:
3603:
3579:
3555:
3533:
3480:
3285:
3138:
3105:
3065:
2977:
2955:
2836:
2761:
2663:contain an element whose square is −1 (namely the
2647:
2569:
2490:
2450:
2425:
2175:
2150:
2112:
1839:
1799:
1771:
1674:
1629:
1587:
1555:
1523:
1481:
1449:
1427:
1405:
1380:
1158:
1094:
1039:
887:
856:
782:
754:
726:
698:
670:
616:
561:
530:
463:
392:
354:
334:
307:
279:
259:
239:
210:
187:
82:
53:
5090:(CSAs) – ring extensions over a field, which are
4555:{\displaystyle \mathbb {Q} (x,y)/\mathbb {Q} (x)}
4473:{\displaystyle \mathbb {Q} (x,y)/\mathbb {Q} (x)}
5411:
5367:Introduction To Modern Algebra, Revised Edition
5137:defined over the field, such as polynomials or
4090:{\displaystyle \mathbb {Q} (x,y)/\mathbb {Q} ,}
4864:if the minimal polynomial of every element of
3653:an isomorphism the largest extension field of
3383:is also finite, as well as the sub extensions
2762:{\displaystyle GF(p^{n})=\mathbb {F} _{p^{n}}}
1595:is finite. This is a simple extension because
1630:{\displaystyle \mathbb {C} =\mathbb {R} (i).}
915:. It is the intersection of all subfields of
4781:completely factors into linear factors over
4575:
4569:
4493:
4487:
4389:
4383:
4273:
4247:
2773:elements; this is an extension field of the
1840:{\displaystyle \left\{1,{\sqrt {2}}\right\}}
1550:
1538:
1147:
1115:
1034:
1002:
3782:
3488:is an extension such that every element of
1159:{\displaystyle K(\{x_{1},\ldots ,x_{n}\}),}
3286:{\displaystyle 1,s,s^{2},\ldots ,s^{d-1},}
2963:It is a transcendental extension field of
647:is an extension that has a finite degree.
635:. Extensions of degree 2 and 3 are called
129:. For example, under the usual notions of
4539:
4514:
4457:
4432:
4410:
4220:
4125:
4080:
4055:
3765:
3739:
3717:
3695:
3673:
3619:
3549:
3504:
3308:form a subextension, which is called the
3297:is the degree of the minimal polynomial.
2971:
2937:
2830:
2817:
2803:
2742:
2478:
2444:
2407:
2233:
2198:
2166:
2151:{\displaystyle \mathbb {Q} ({\sqrt {2}})}
2131:
2094:
1985:
1904:
1865:
1790:
1757:
1700:
1655:
1647:
1611:
1603:
1588:{\displaystyle \mathbb {C} /\mathbb {R} }
1581:
1571:
1508:
1500:
1482:{\displaystyle \mathbb {C} /\mathbb {Q} }
1475:
1465:
1443:
1421:
1399:
1374:
1040:{\displaystyle S=\{x_{1},\ldots ,x_{n}\}}
5317:
5295:
5280:
5268:
5256:
5244:
5232:
5208:
5196:
5058:between the intermediate fields and the
1347:, so field extensions are precisely the
5114:
5074:Field extensions can be generalized to
4733:Normal, separable and Galois extensions
5412:
5086:. A closer non-commutative analog are
5062:of the Galois group, described by the
3857:. It is always possible to find a set
3026:
1388:is an extension field of the field of
1095:{\displaystyle K(x_{1},\ldots ,x_{n})}
5364:
5220:
5339:
5064:fundamental theorem of Galois theory
4212:Such an extension can be defined as
4047:For example, consider the extension
3197:if and only if the simple extension
443:. Such a field extension is denoted
148:Field extensions are fundamental in
3649:has an algebraic closure, which is
1667:
1489:is also a field extension. We have
585:of this vector space is called the
490:, which is in turn an extension of
13:
5298:A First Course In Abstract Algebra
5147:extension of scalars: applications
5125:Given a field extension, one can "
5069:
3709:, but not an algebraic closure of
1686:), so this extension is infinite.
903:, there is a smallest subfield of
790:are finite. In this case, one has
410:
14:
5431:
5385:
4957:{\displaystyle {\text{Aut}}(L/K)}
4797:, which is an extension field of
3861:, algebraically independent over
1232:is often said to result from the
4714:generates the whole extension.
3638:A simple extension is algebraic
2956:{\displaystyle \mathbb {C} (M).}
2589:generated by this polynomial is
2491:{\displaystyle \mathbb {Q} _{p}}
1675:{\displaystyle ={\mathfrak {c}}}
1284:) is isomorphic to the field of
322:, the latter definition implies
3665:has a root in it. For example,
1221:An extension field of the form
439:, and this pair of fields is a
377:to) a subfield of any field of
5274:
5262:
5250:
5238:
5226:
5214:
5202:
5190:
4951:
4937:
4549:
4543:
4530:
4518:
4467:
4461:
4448:
4436:
4239:
4233:
4230:
4224:
4205:{\displaystyle y^{2}-x^{3}=0.}
4071:
4059:
3731:, as it is not algebraic over
3528:
3508:
3328:are algebraic, the extensions
2947:
2941:
2911:. This extension is infinite.
2795:
2789:
2734:
2721:
2642:
2623:
2615:
2609:
2370:
2349:
2334:
2313:
2304:
2284:
2257:
2237:
2222:
2202:
2145:
2135:
2123:is an extension field of both
1714:
1704:
1659:
1643:
1621:
1615:
1512:
1496:
1150:
1112:
1089:
1057:
848:
836:
830:
818:
812:
800:
734:is finite if and only if both
611:
599:
141:are an extension field of the
90:, such that the operations of
1:
5349:Graduate Texts in Mathematics
5289:
4376:Obviously, the singleton set
4135:{\displaystyle \mathbb {Q} ,}
4012:is purely transcendental and
3841:exists among the elements of
3629:{\displaystyle \mathbb {Q} .}
3541:is an algebraic extension of
3174:. This minimal polynomial is
2682:. This is an extension field
2648:{\displaystyle L=K/(X^{2}+1)}
2536:does not contain any element
2532:). Suppose for instance that
2176:{\displaystyle \mathbb {Q} ,}
1800:{\displaystyle \mathbb {Q} ,}
1563:is a basis, so the extension
1366:The field of complex numbers
1188:consists of a single element
5324:Blaisdell Publishing Company
5014:For a given field extension
4785:. Every algebraic extension
4417:{\displaystyle \mathbb {Q} }
3772:{\displaystyle \mathbb {Q} }
3746:{\displaystyle \mathbb {Q} }
3724:{\displaystyle \mathbb {Q} }
3702:{\displaystyle \mathbb {R} }
3680:{\displaystyle \mathbb {C} }
3556:{\displaystyle \mathbb {Q} }
2978:{\displaystyle \mathbb {C} }
2855:, we can consider the field
2451:{\displaystyle \mathbb {Q} }
1684:cardinality of the continuum
1450:{\displaystyle \mathbb {Q} }
1428:{\displaystyle \mathbb {R} }
1406:{\displaystyle \mathbb {R} }
1381:{\displaystyle \mathbb {C} }
362:have the same zero element.
240:{\displaystyle K\subseteq L}
83:{\displaystyle K\subseteq L}
7:
5398:Encyclopedia of Mathematics
5152:
4725:. The problem of finding a
4001:are algebraically closed.
3877:) is algebraic. Such a set
3687:is an algebraic closure of
3604:{\displaystyle {\sqrt {3}}}
3580:{\displaystyle {\sqrt {2}}}
3300:The set of the elements of
3106:{\displaystyle {\sqrt {2}}}
3019:, is an extension field of
2993:. More generally, given an
1361:
167:
10:
5436:
5296:Fraleigh, John B. (1976),
5118:
4964:, consisting of all field
4894:Given any field extension
3786:
3162:of lowest degree that has
3030:
2544:= −1. Then the polynomial
1299:
365:For example, the field of
5300:(2nd ed.), Reading:
4889:primitive element theorem
3831:algebraically independent
2655:is an extension field of
1782:is an extension field of
1276:is not finite, the field
1257:primitive element theorem
160:, and are widely used in
5318:Herstein, I. N. (1964),
5184:
5164:Glossary of field theory
4793:admits a normal closure
4727:rational parametrization
4687:{\displaystyle y=t^{3},}
3981:are transcendental over
3793:Given a field extension
3789:Transcendental extension
3783:Transcendental extension
3657:which is algebraic over
3304:that are algebraic over
3166:as a root is called the
3139:{\displaystyle x^{2}-2.}
867:Given a field extension
857:{\displaystyle =\cdot .}
541:Given a field extension
295:of a nonzero element of
5365:McCoy, Neal H. (1968),
5088:central simple algebras
4832:An algebraic extension
4737:An algebraic extension
4651:{\displaystyle x=t^{2}}
4402:is transcendental over
4117:is transcendental over
2929:is a field, denoted by
2903:of the polynomial ring
2883:) are fractions of two
2678:of any polynomial from
2570:{\displaystyle X^{2}+1}
2524:for a given polynomial
2520:in order to "create" a
2460:algebraic number fields
1847:can serve as a basis.
1556:{\displaystyle \{1,i\}}
150:algebraic number theory
5393:"Extension of a field"
5036:
4958:
4922:, we can consider its
4916:
4854:
4823:
4771:irreducible polynomial
4759:
4708:
4688:
4652:
4619:
4618:{\displaystyle t=y/x,}
4582:
4556:
4500:
4474:
4418:
4396:
4370:
4347:
4323:
4303:
4283:
4206:
4156:
4136:
4111:
4091:
3955:
3915:
3815:
3773:
3757:is not algebraic over
3747:
3725:
3703:
3681:
3630:
3605:
3581:
3557:
3535:
3482:
3287:
3140:
3107:
3067:
2979:
2957:
2838:
2763:
2649:
2571:
2492:
2452:
2427:
2177:
2152:
2114:
1841:
1801:
1773:
1676:
1631:
1589:
1557:
1525:
1483:
1451:
1429:
1407:
1382:
1262:If a simple extension
1160:
1096:
1047:is finite, one writes
1041:
889:
858:
784:
756:
728:
700:
672:
618:
563:
532:
504:intermediate extension
465:
394:
356:
336:
309:
281:
261:
241:
212:
189:
152:, and in the study of
84:
55:
5143:group representations
5037:
4959:
4917:
4887:A consequence of the
4855:
4824:
4760:
4709:
4689:
4653:
4620:
4583:
4581:{\displaystyle \{y\}}
4557:
4501:
4499:{\displaystyle \{x\}}
4475:
4419:
4397:
4395:{\displaystyle \{x\}}
4371:
4348:
4324:
4304:
4284:
4207:
4157:
4137:
4112:
4092:
3956:
3925:purely transcendental
3916:
3816:
3774:
3748:
3726:
3704:
3682:
3631:
3606:
3582:
3558:
3536:
3483:
3288:
3141:
3108:
3085:with coefficients in
3068:
3045:of a field extension
2980:
2958:
2923:meromorphic functions
2871:with coefficients in
2839:
2764:
2650:
2572:
2493:
2462:and are important in
2453:
2436:Finite extensions of
2428:
2178:
2153:
2115:
1842:
1802:
1774:
1677:
1632:
1590:
1558:
1526:
1484:
1452:
1430:
1408:
1383:
1161:
1097:
1042:
890:
859:
785:
757:
729:
701:
673:
650:Given two extensions
619:
564:
533:
466:
395:
369:is a subfield of the
357:
337:
310:
282:
262:
242:
213:
190:
85:
56:
5135:associative algebras
5121:Extension of scalars
5115:Extension of scalars
5018:
4929:
4898:
4836:
4805:
4741:
4698:
4662:
4629:
4592:
4566:
4510:
4484:
4480:is algebraic; hence
4428:
4406:
4380:
4357:
4337:
4313:
4293:
4216:
4170:
4146:
4121:
4101:
4051:
3937:
3897:
3847:transcendence degree
3797:
3761:
3735:
3713:
3691:
3669:
3615:
3591:
3567:
3545:
3500:
3464:
3230:
3117:
3093:
3049:
2967:
2933:
2780:
2712:
2597:
2548:
2473:
2440:
2190:
2162:
2127:
1857:
1811:
1786:
1696:
1640:
1599:
1567:
1535:
1493:
1461:
1439:
1417:
1395:
1370:
1339:between two fields.
1106:
1051:
993:
927:, and is denoted by
871:
797:
766:
738:
710:
682:
654:
637:quadratic extensions
596:
545:
514:
447:
384:
346:
326:
299:
271:
251:
225:
202:
179:
68:
37:
5141:and the associated
5078:which consist of a
5035:{\displaystyle L/K}
4915:{\displaystyle L/K}
4853:{\displaystyle L/K}
4822:{\displaystyle L/K}
4777:that has a root in
4758:{\displaystyle L/K}
4331:equivalence classes
3954:{\displaystyle L/K}
3914:{\displaystyle L/K}
3883:transcendence basis
3814:{\displaystyle L/K}
3611:are algebraic over
3481:{\displaystyle L/K}
3459:algebraic extension
3454:are all algebraic.
3436:). It follows that
3066:{\displaystyle L/K}
3033:Algebraic extension
3027:Algebraic extension
2585:, consequently the
2498:for a prime number
888:{\displaystyle L/K}
783:{\displaystyle M/L}
755:{\displaystyle L/K}
727:{\displaystyle M/K}
699:{\displaystyle M/L}
671:{\displaystyle L/K}
569:, the larger field
562:{\displaystyle L/K}
531:{\displaystyle L/K}
486:is an extension of
464:{\displaystyle L/K}
54:{\displaystyle L/K}
5096:center of the ring
5032:
5009:abelian extensions
4954:
4924:automorphism group
4912:
4850:
4819:
4755:
4723:rational varieties
4704:
4684:
4648:
4615:
4578:
4552:
4496:
4470:
4424:and the extension
4414:
4392:
4369:{\displaystyle Y.}
4366:
4343:
4319:
4299:
4279:
4202:
4152:
4132:
4107:
4087:
3951:
3911:
3811:
3769:
3743:
3721:
3699:
3677:
3626:
3601:
3577:
3553:
3531:
3492:is algebraic over
3478:
3283:
3193:is algebraic over
3168:minimal polynomial
3154:is algebraic over
3136:
3103:
3073:is algebraic over
3063:
2975:
2953:
2901:field of fractions
2875:; the elements of
2865:rational functions
2834:
2759:
2645:
2567:
2488:
2448:
2423:
2421:
2173:
2148:
2110:
2108:
1837:
1797:
1769:
1672:
1627:
1585:
1553:
1524:{\displaystyle =2}
1521:
1479:
1447:
1425:
1403:
1378:
1353:category of fields
1286:rational fractions
1218:of the extension.
1177:finitely generated
1166:and one says that
1156:
1092:
1037:
949:"). One says that
885:
854:
780:
752:
724:
696:
668:
643:, respectively. A
614:
592:and is denoted by
559:
528:
500:intermediate field
461:
390:
352:
332:
305:
277:
257:
237:
208:
185:
162:algebraic geometry
80:
51:
25:, particularly in
5358:978-0-387-95385-4
5320:Topics In Algebra
5179:Regular extension
5174:Primary extension
5101:Brauer equivalent
4935:
4707:{\displaystyle t}
4346:{\displaystyle X}
4322:{\displaystyle y}
4302:{\displaystyle x}
4155:{\displaystyle y}
4110:{\displaystyle x}
3599:
3575:
3526:
3516:
3365:are finite. Thus
3310:algebraic closure
3101:
3037:Algebraic element
2995:algebraic variety
2987:constant function
2921:, the set of all
2367:
2357:
2331:
2321:
2302:
2292:
2255:
2245:
2220:
2210:
2143:
2064:
2051:
2038:
1998:
1967:
1932:
1917:
1889:
1879:
1830:
1739:
1712:
1337:ring homomorphism
1216:primitive element
631:trivial extension
498:is said to be an
419:is a subfield of
393:{\displaystyle 0}
355:{\displaystyle L}
335:{\displaystyle K}
308:{\displaystyle K}
280:{\displaystyle 1}
260:{\displaystyle L}
211:{\displaystyle L}
188:{\displaystyle K}
5427:
5420:Field extensions
5406:
5381:
5361:
5336:
5314:
5284:
5278:
5272:
5266:
5260:
5254:
5248:
5242:
5236:
5230:
5224:
5218:
5212:
5206:
5200:
5194:
5131:complexification
5105:Azumaya algebras
5041:
5039:
5038:
5033:
5028:
4963:
4961:
4960:
4955:
4947:
4936:
4933:
4921:
4919:
4918:
4913:
4908:
4882:Galois extension
4859:
4857:
4856:
4851:
4846:
4828:
4826:
4825:
4820:
4815:
4764:
4762:
4761:
4756:
4751:
4713:
4711:
4710:
4705:
4693:
4691:
4690:
4685:
4680:
4679:
4657:
4655:
4654:
4649:
4647:
4646:
4624:
4622:
4621:
4616:
4608:
4587:
4585:
4584:
4579:
4561:
4559:
4558:
4553:
4542:
4537:
4517:
4505:
4503:
4502:
4497:
4479:
4477:
4476:
4471:
4460:
4455:
4435:
4423:
4421:
4420:
4415:
4413:
4401:
4399:
4398:
4393:
4375:
4373:
4372:
4367:
4352:
4350:
4349:
4344:
4328:
4326:
4325:
4320:
4308:
4306:
4305:
4300:
4288:
4286:
4285:
4280:
4272:
4271:
4259:
4258:
4246:
4223:
4211:
4209:
4208:
4203:
4195:
4194:
4182:
4181:
4166:of the equation
4161:
4159:
4158:
4153:
4141:
4139:
4138:
4133:
4128:
4116:
4114:
4113:
4108:
4096:
4094:
4093:
4088:
4083:
4078:
4058:
3977:except those of
3960:
3958:
3957:
3952:
3947:
3927:
3926:
3920:
3918:
3917:
3912:
3907:
3820:
3818:
3817:
3812:
3807:
3778:
3776:
3775:
3770:
3768:
3756:
3752:
3750:
3749:
3744:
3742:
3730:
3728:
3727:
3722:
3720:
3708:
3706:
3705:
3700:
3698:
3686:
3684:
3683:
3678:
3676:
3635:
3633:
3632:
3627:
3622:
3610:
3608:
3607:
3602:
3600:
3595:
3586:
3584:
3583:
3578:
3576:
3571:
3562:
3560:
3559:
3554:
3552:
3540:
3538:
3537:
3532:
3527:
3522:
3517:
3512:
3507:
3487:
3485:
3484:
3479:
3474:
3445:
3435:
3428:
3414:
3400:
3382:
3364:
3341:
3292:
3290:
3289:
3284:
3279:
3278:
3254:
3253:
3210:
3160:monic polynomial
3145:
3143:
3142:
3137:
3129:
3128:
3112:
3110:
3109:
3104:
3102:
3097:
3072:
3070:
3069:
3064:
3059:
3000:over some field
2984:
2982:
2981:
2976:
2974:
2962:
2960:
2959:
2954:
2940:
2867:in the variable
2843:
2841:
2840:
2835:
2833:
2825:
2820:
2812:
2811:
2806:
2768:
2766:
2765:
2760:
2758:
2757:
2756:
2755:
2745:
2733:
2732:
2654:
2652:
2651:
2646:
2635:
2634:
2622:
2576:
2574:
2573:
2568:
2560:
2559:
2497:
2495:
2494:
2489:
2487:
2486:
2481:
2458:are also called
2457:
2455:
2454:
2449:
2447:
2432:
2430:
2429:
2424:
2422:
2415:
2411:
2410:
2378:
2377:
2368:
2363:
2358:
2353:
2342:
2341:
2332:
2327:
2322:
2317:
2303:
2298:
2293:
2288:
2263:
2256:
2251:
2246:
2241:
2236:
2221:
2216:
2211:
2206:
2201:
2182:
2180:
2179:
2174:
2169:
2157:
2155:
2154:
2149:
2144:
2139:
2134:
2119:
2117:
2116:
2111:
2109:
2102:
2098:
2097:
2065:
2060:
2052:
2047:
2039:
2034:
2012:
2008:
2004:
2003:
1999:
1994:
1988:
1968:
1963:
1941:
1937:
1933:
1928:
1922:
1918:
1913:
1907:
1895:
1891:
1890:
1885:
1880:
1875:
1868:
1846:
1844:
1843:
1838:
1836:
1832:
1831:
1826:
1806:
1804:
1803:
1798:
1793:
1778:
1776:
1775:
1770:
1765:
1761:
1760:
1740:
1735:
1713:
1708:
1703:
1681:
1679:
1678:
1673:
1671:
1670:
1658:
1650:
1636:
1634:
1633:
1628:
1614:
1606:
1594:
1592:
1591:
1586:
1584:
1579:
1574:
1562:
1560:
1559:
1554:
1530:
1528:
1527:
1522:
1511:
1503:
1488:
1486:
1485:
1480:
1478:
1473:
1468:
1457:. Clearly then,
1456:
1454:
1453:
1448:
1446:
1434:
1432:
1431:
1426:
1424:
1412:
1410:
1409:
1404:
1402:
1387:
1385:
1384:
1379:
1377:
1275:
1238:
1237:
1231:
1208:simple extension
1205:
1192:, the extension
1179:
1178:
1165:
1163:
1162:
1157:
1146:
1145:
1127:
1126:
1101:
1099:
1098:
1093:
1088:
1087:
1069:
1068:
1046:
1044:
1043:
1038:
1033:
1032:
1014:
1013:
944:
943:
894:
892:
891:
886:
881:
863:
861:
860:
855:
789:
787:
786:
781:
776:
761:
759:
758:
753:
748:
733:
731:
730:
725:
720:
706:, the extension
705:
703:
702:
697:
692:
677:
675:
674:
669:
664:
645:finite extension
641:cubic extensions
633:
632:
623:
621:
620:
617:{\displaystyle }
615:
590:of the extension
568:
566:
565:
560:
555:
537:
535:
534:
529:
524:
470:
468:
467:
462:
457:
399:
397:
396:
391:
367:rational numbers
361:
359:
358:
353:
341:
339:
338:
333:
321:
314:
312:
311:
306:
286:
284:
283:
278:
266:
264:
263:
258:
246:
244:
243:
238:
217:
215:
214:
209:
194:
192:
191:
186:
154:polynomial roots
105:. In this case,
89:
87:
86:
81:
60:
58:
57:
52:
47:
5435:
5434:
5430:
5429:
5428:
5426:
5425:
5424:
5410:
5409:
5391:
5388:
5371:Allyn and Bacon
5359:
5334:
5312:
5292:
5287:
5279:
5275:
5267:
5263:
5255:
5251:
5243:
5239:
5231:
5227:
5219:
5215:
5207:
5203:
5195:
5191:
5187:
5169:Tower of fields
5155:
5123:
5117:
5082:and one of its
5076:ring extensions
5072:
5070:Generalizations
5024:
5019:
5016:
5015:
4943:
4932:
4930:
4927:
4926:
4904:
4899:
4896:
4895:
4842:
4837:
4834:
4833:
4811:
4806:
4803:
4802:
4747:
4742:
4739:
4738:
4735:
4719:function fields
4699:
4696:
4695:
4675:
4671:
4663:
4660:
4659:
4642:
4638:
4630:
4627:
4626:
4604:
4593:
4590:
4589:
4567:
4564:
4563:
4538:
4533:
4513:
4511:
4508:
4507:
4485:
4482:
4481:
4456:
4451:
4431:
4429:
4426:
4425:
4409:
4407:
4404:
4403:
4381:
4378:
4377:
4358:
4355:
4354:
4338:
4335:
4334:
4314:
4311:
4310:
4294:
4291:
4290:
4267:
4263:
4254:
4250:
4242:
4219:
4217:
4214:
4213:
4190:
4186:
4177:
4173:
4171:
4168:
4167:
4147:
4144:
4143:
4124:
4122:
4119:
4118:
4102:
4099:
4098:
4079:
4074:
4054:
4052:
4049:
4048:
3943:
3938:
3935:
3934:
3924:
3923:
3903:
3898:
3895:
3894:
3803:
3798:
3795:
3794:
3791:
3785:
3764:
3762:
3759:
3758:
3754:
3738:
3736:
3733:
3732:
3716:
3714:
3711:
3710:
3694:
3692:
3689:
3688:
3672:
3670:
3667:
3666:
3618:
3616:
3613:
3612:
3594:
3592:
3589:
3588:
3570:
3568:
3565:
3564:
3548:
3546:
3543:
3542:
3521:
3511:
3503:
3501:
3498:
3497:
3470:
3465:
3462:
3461:
3437:
3430:
3416:
3402:
3384:
3366:
3343:
3329:
3268:
3264:
3249:
3245:
3231:
3228:
3227:
3198:
3124:
3120:
3118:
3115:
3114:
3096:
3094:
3091:
3090:
3089:. For example,
3055:
3050:
3047:
3046:
3039:
3031:Main articles:
3029:
2970:
2968:
2965:
2964:
2936:
2934:
2931:
2930:
2916:Riemann surface
2829:
2821:
2816:
2807:
2802:
2801:
2781:
2778:
2777:
2751:
2747:
2746:
2741:
2740:
2728:
2724:
2713:
2710:
2709:
2676:splitting field
2630:
2626:
2618:
2598:
2595:
2594:
2555:
2551:
2549:
2546:
2545:
2515:polynomial ring
2482:
2477:
2476:
2474:
2471:
2470:
2443:
2441:
2438:
2437:
2420:
2419:
2406:
2373:
2369:
2362:
2352:
2337:
2333:
2326:
2316:
2297:
2287:
2274:
2270:
2261:
2260:
2250:
2240:
2232:
2225:
2215:
2205:
2197:
2193:
2191:
2188:
2187:
2165:
2163:
2160:
2159:
2138:
2130:
2128:
2125:
2124:
2107:
2106:
2093:
2059:
2046:
2033:
2023:
2019:
2010:
2009:
1993:
1989:
1984:
1962:
1952:
1948:
1939:
1938:
1927:
1923:
1912:
1908:
1903:
1896:
1884:
1874:
1873:
1869:
1864:
1860:
1858:
1855:
1854:
1825:
1818:
1814:
1812:
1809:
1808:
1789:
1787:
1784:
1783:
1756:
1734:
1724:
1720:
1707:
1699:
1697:
1694:
1693:
1666:
1665:
1654:
1646:
1641:
1638:
1637:
1610:
1602:
1600:
1597:
1596:
1580:
1575:
1570:
1568:
1565:
1564:
1536:
1533:
1532:
1507:
1499:
1494:
1491:
1490:
1474:
1469:
1464:
1462:
1459:
1458:
1442:
1440:
1437:
1436:
1420:
1418:
1415:
1414:
1398:
1396:
1393:
1392:
1373:
1371:
1368:
1367:
1364:
1302:
1263:
1235:
1234:
1222:
1193:
1176:
1175:
1141:
1137:
1122:
1118:
1107:
1104:
1103:
1083:
1079:
1064:
1060:
1052:
1049:
1048:
1028:
1024:
1009:
1005:
994:
991:
990:
957:) is the field
941:
940:
877:
872:
869:
868:
798:
795:
794:
772:
767:
764:
763:
744:
739:
736:
735:
716:
711:
708:
707:
688:
683:
680:
679:
660:
655:
652:
651:
630:
629:
597:
594:
593:
551:
546:
543:
542:
520:
515:
512:
511:
453:
448:
445:
444:
441:field extension
429:extension field
413:
411:Extension field
385:
382:
381:
347:
344:
343:
327:
324:
323:
319:
300:
297:
296:
272:
269:
268:
252:
249:
248:
226:
223:
222:
203:
200:
199:
180:
177:
176:
170:
139:complex numbers
111:extension field
69:
66:
65:
61:) is a pair of
43:
38:
35:
34:
31:field extension
19:
12:
11:
5:
5433:
5423:
5422:
5408:
5407:
5387:
5386:External links
5384:
5383:
5382:
5362:
5357:
5337:
5333:978-1114541016
5332:
5315:
5310:
5302:Addison-Wesley
5291:
5288:
5286:
5285:
5283:, p. 169)
5281:Herstein (1964
5273:
5271:, p. 319)
5269:Fraleigh (1976
5261:
5259:, p. 363)
5257:Fraleigh (1976
5249:
5247:, p. 193)
5245:Herstein (1964
5237:
5235:, p. 298)
5233:Fraleigh (1976
5225:
5223:, p. 116)
5213:
5211:, p. 167)
5209:Herstein (1964
5201:
5199:, p. 293)
5197:Fraleigh (1976
5188:
5186:
5183:
5182:
5181:
5176:
5171:
5166:
5161:
5154:
5151:
5139:group algebras
5127:extend scalars
5119:Main article:
5116:
5113:
5092:simple algebra
5071:
5068:
5046:(subfields of
5031:
5027:
5023:
4953:
4950:
4946:
4942:
4939:
4911:
4907:
4903:
4849:
4845:
4841:
4818:
4814:
4810:
4754:
4750:
4746:
4734:
4731:
4703:
4683:
4678:
4674:
4670:
4667:
4645:
4641:
4637:
4634:
4614:
4611:
4607:
4603:
4600:
4597:
4577:
4574:
4571:
4551:
4548:
4545:
4541:
4536:
4532:
4529:
4526:
4523:
4520:
4516:
4495:
4492:
4489:
4469:
4466:
4463:
4459:
4454:
4450:
4447:
4444:
4441:
4438:
4434:
4412:
4391:
4388:
4385:
4365:
4362:
4342:
4318:
4298:
4278:
4275:
4270:
4266:
4262:
4257:
4253:
4249:
4245:
4241:
4238:
4235:
4232:
4229:
4226:
4222:
4201:
4198:
4193:
4189:
4185:
4180:
4176:
4151:
4131:
4127:
4106:
4086:
4082:
4077:
4073:
4070:
4067:
4064:
4061:
4057:
3950:
3946:
3942:
3921:is said to be
3910:
3906:
3902:
3810:
3806:
3802:
3787:Main article:
3784:
3781:
3767:
3741:
3719:
3697:
3675:
3640:if and only if
3625:
3621:
3598:
3574:
3551:
3530:
3525:
3520:
3515:
3510:
3506:
3477:
3473:
3469:
3282:
3277:
3274:
3271:
3267:
3263:
3260:
3257:
3252:
3248:
3244:
3241:
3238:
3235:
3226:) consists of
3146:If an element
3135:
3132:
3127:
3123:
3100:
3062:
3058:
3054:
3028:
3025:
3006:function field
2973:
2952:
2949:
2946:
2943:
2939:
2851:Given a field
2832:
2828:
2824:
2819:
2815:
2810:
2805:
2800:
2797:
2794:
2791:
2788:
2785:
2754:
2750:
2744:
2739:
2736:
2731:
2727:
2723:
2720:
2717:
2644:
2641:
2638:
2633:
2629:
2625:
2621:
2617:
2614:
2611:
2608:
2605:
2602:
2566:
2563:
2558:
2554:
2485:
2480:
2468:p-adic numbers
2446:
2434:
2433:
2418:
2414:
2409:
2405:
2402:
2399:
2396:
2393:
2390:
2387:
2384:
2381:
2376:
2372:
2366:
2361:
2356:
2351:
2348:
2345:
2340:
2336:
2330:
2325:
2320:
2315:
2312:
2309:
2306:
2301:
2296:
2291:
2286:
2283:
2280:
2277:
2273:
2269:
2266:
2264:
2262:
2259:
2254:
2249:
2244:
2239:
2235:
2231:
2228:
2226:
2224:
2219:
2214:
2209:
2204:
2200:
2196:
2195:
2172:
2168:
2147:
2142:
2137:
2133:
2121:
2120:
2105:
2101:
2096:
2092:
2089:
2086:
2083:
2080:
2077:
2074:
2071:
2068:
2063:
2058:
2055:
2050:
2045:
2042:
2037:
2032:
2029:
2026:
2022:
2018:
2015:
2013:
2011:
2007:
2002:
1997:
1992:
1987:
1983:
1980:
1977:
1974:
1971:
1966:
1961:
1958:
1955:
1951:
1947:
1944:
1942:
1940:
1936:
1931:
1926:
1921:
1916:
1911:
1906:
1902:
1899:
1897:
1894:
1888:
1883:
1878:
1872:
1867:
1863:
1862:
1835:
1829:
1824:
1821:
1817:
1796:
1792:
1780:
1779:
1768:
1764:
1759:
1755:
1752:
1749:
1746:
1743:
1738:
1733:
1730:
1727:
1723:
1719:
1716:
1711:
1706:
1702:
1669:
1664:
1661:
1657:
1653:
1649:
1645:
1626:
1623:
1620:
1617:
1613:
1609:
1605:
1583:
1578:
1573:
1552:
1549:
1546:
1543:
1540:
1520:
1517:
1514:
1510:
1506:
1502:
1498:
1477:
1472:
1467:
1445:
1423:
1401:
1376:
1363:
1360:
1318:quotient group
1301:
1298:
1253:characteristic
1155:
1152:
1149:
1144:
1140:
1136:
1133:
1130:
1125:
1121:
1117:
1114:
1111:
1091:
1086:
1082:
1078:
1075:
1072:
1067:
1063:
1059:
1056:
1036:
1031:
1027:
1023:
1020:
1017:
1012:
1008:
1004:
1001:
998:
975:generating set
907:that contains
884:
880:
876:
865:
864:
853:
850:
847:
844:
841:
838:
835:
832:
829:
826:
823:
820:
817:
814:
811:
808:
805:
802:
779:
775:
771:
751:
747:
743:
723:
719:
715:
695:
691:
687:
667:
663:
659:
613:
610:
607:
604:
601:
558:
554:
550:
527:
523:
519:
460:
456:
452:
412:
409:
405:characteristic
389:
379:characteristic
351:
331:
304:
276:
256:
236:
233:
230:
207:
184:
169:
166:
135:multiplication
79:
76:
73:
50:
46:
42:
17:
9:
6:
4:
3:
2:
5432:
5421:
5418:
5417:
5415:
5404:
5400:
5399:
5394:
5390:
5389:
5380:
5376:
5372:
5368:
5363:
5360:
5354:
5350:
5346:
5342:
5338:
5335:
5329:
5325:
5321:
5316:
5313:
5311:0-201-01984-1
5307:
5303:
5299:
5294:
5293:
5282:
5277:
5270:
5265:
5258:
5253:
5246:
5241:
5234:
5229:
5222:
5217:
5210:
5205:
5198:
5193:
5189:
5180:
5177:
5175:
5172:
5170:
5167:
5165:
5162:
5160:
5157:
5156:
5150:
5148:
5144:
5140:
5136:
5132:
5128:
5122:
5112:
5110:
5106:
5102:
5097:
5093:
5089:
5085:
5081:
5077:
5067:
5065:
5061:
5057:
5053:
5050:that contain
5049:
5045:
5029:
5025:
5021:
5012:
5010:
5006:
5002:
4998:
4994:
4990:
4986:
4982:
4978:
4974:
4970:
4967:
4966:automorphisms
4948:
4944:
4940:
4925:
4909:
4905:
4901:
4892:
4890:
4885:
4883:
4879:
4875:
4871:
4867:
4863:
4847:
4843:
4839:
4830:
4816:
4812:
4808:
4800:
4796:
4792:
4788:
4784:
4780:
4776:
4772:
4768:
4752:
4748:
4744:
4730:
4728:
4724:
4720:
4715:
4701:
4681:
4676:
4672:
4668:
4665:
4643:
4639:
4635:
4632:
4612:
4609:
4605:
4601:
4598:
4595:
4572:
4562:. Similarly,
4546:
4534:
4527:
4524:
4521:
4490:
4464:
4452:
4445:
4442:
4439:
4386:
4363:
4360:
4340:
4332:
4316:
4296:
4276:
4268:
4264:
4260:
4255:
4251:
4243:
4236:
4227:
4199:
4196:
4191:
4187:
4183:
4178:
4174:
4165:
4149:
4129:
4104:
4084:
4075:
4068:
4065:
4062:
4045:
4043:
4039:
4035:
4031:
4027:
4023:
4019:
4015:
4011:
4007:
4002:
4000:
3996:
3992:
3988:
3984:
3980:
3976:
3972:
3968:
3964:
3948:
3944:
3940:
3932:
3928:
3908:
3904:
3900:
3892:
3888:
3884:
3880:
3876:
3872:
3868:
3864:
3860:
3856:
3852:
3848:
3844:
3840:
3836:
3832:
3828:
3824:
3808:
3804:
3800:
3790:
3780:
3753:(for example
3664:
3660:
3656:
3652:
3648:
3643:
3641:
3636:
3623:
3596:
3572:
3523:
3518:
3513:
3495:
3491:
3475:
3471:
3467:
3460:
3455:
3453:
3449:
3444:
3440:
3433:
3427:
3423:
3419:
3413:
3409:
3405:
3399:
3395:
3391:
3387:
3381:
3377:
3373:
3369:
3362:
3358:
3354:
3350:
3346:
3340:
3336:
3332:
3327:
3323:
3319:
3315:
3311:
3307:
3303:
3298:
3296:
3280:
3275:
3272:
3269:
3265:
3261:
3258:
3255:
3250:
3246:
3242:
3239:
3236:
3233:
3225:
3221:
3218:
3214:
3209:
3205:
3201:
3196:
3192:
3188:
3183:
3181:
3177:
3173:
3169:
3165:
3161:
3157:
3153:
3149:
3133:
3130:
3125:
3121:
3098:
3088:
3084:
3081:of a nonzero
3080:
3076:
3060:
3056:
3052:
3044:
3038:
3034:
3024:
3022:
3018:
3014:
3010:
3007:
3003:
2999:
2996:
2992:
2988:
2950:
2944:
2928:
2924:
2920:
2917:
2912:
2910:
2906:
2902:
2898:
2894:
2891:, and indeed
2890:
2886:
2882:
2878:
2874:
2870:
2866:
2862:
2858:
2854:
2849:
2847:
2826:
2822:
2813:
2808:
2798:
2792:
2786:
2783:
2776:
2772:
2752:
2748:
2737:
2729:
2725:
2718:
2715:
2708:
2704:
2700:
2696:
2691:
2689:
2685:
2681:
2677:
2672:
2670:
2666:
2665:residue class
2662:
2658:
2639:
2636:
2631:
2627:
2619:
2612:
2606:
2603:
2600:
2592:
2588:
2584:
2580:
2564:
2561:
2556:
2552:
2543:
2539:
2535:
2531:
2527:
2523:
2519:
2516:
2512:
2511:quotient ring
2508:
2503:
2501:
2483:
2469:
2465:
2464:number theory
2461:
2416:
2412:
2403:
2400:
2397:
2394:
2391:
2388:
2385:
2382:
2379:
2374:
2364:
2359:
2354:
2346:
2343:
2338:
2328:
2323:
2318:
2310:
2307:
2299:
2294:
2289:
2281:
2278:
2275:
2271:
2267:
2265:
2252:
2247:
2242:
2229:
2227:
2217:
2212:
2207:
2186:
2185:
2184:
2170:
2140:
2103:
2099:
2090:
2087:
2084:
2081:
2078:
2075:
2072:
2069:
2066:
2061:
2056:
2053:
2048:
2043:
2040:
2035:
2030:
2027:
2024:
2020:
2016:
2014:
2005:
2000:
1995:
1990:
1981:
1978:
1975:
1972:
1969:
1964:
1959:
1956:
1953:
1949:
1945:
1943:
1934:
1929:
1924:
1919:
1914:
1909:
1900:
1898:
1892:
1886:
1881:
1876:
1870:
1853:
1852:
1851:
1848:
1833:
1827:
1822:
1819:
1815:
1794:
1766:
1762:
1753:
1750:
1747:
1744:
1741:
1736:
1731:
1728:
1725:
1721:
1717:
1709:
1692:
1691:
1690:
1687:
1685:
1662:
1651:
1624:
1618:
1607:
1576:
1547:
1544:
1541:
1518:
1515:
1504:
1470:
1391:
1359:
1356:
1354:
1350:
1346:
1342:
1338:
1335:
1329:
1327:
1323:
1319:
1315:
1314:quotient ring
1311:
1307:
1304:The notation
1297:
1295:
1291:
1287:
1283:
1279:
1274:
1270:
1266:
1260:
1258:
1254:
1249:
1247:
1243:
1239:
1229:
1225:
1219:
1217:
1213:
1209:
1204:
1200:
1196:
1191:
1187:
1183:
1173:
1169:
1153:
1142:
1138:
1134:
1131:
1128:
1123:
1119:
1109:
1084:
1080:
1076:
1073:
1070:
1065:
1061:
1054:
1029:
1025:
1021:
1018:
1015:
1010:
1006:
999:
996:
988:
984:
980:
976:
972:
968:
964:
960:
956:
952:
948:
945:
938:
934:
930:
926:
922:
919:that contain
918:
914:
910:
906:
902:
898:
895:and a subset
882:
878:
874:
851:
845:
842:
839:
833:
827:
824:
821:
815:
809:
806:
803:
793:
792:
791:
777:
773:
769:
749:
745:
741:
721:
717:
713:
693:
689:
685:
665:
661:
657:
648:
646:
642:
638:
634:
625:
608:
605:
602:
591:
589:
584:
580:
576:
572:
556:
552:
548:
539:
525:
521:
517:
509:
505:
501:
497:
493:
489:
485:
480:
478:
474:
458:
454:
450:
442:
438:
434:
430:
426:
422:
418:
408:
406:
401:
387:
380:
376:
372:
368:
363:
349:
329:
316:
302:
294:
290:
274:
254:
234:
231:
228:
221:
205:
198:
182:
175:
165:
163:
159:
158:Galois theory
155:
151:
146:
144:
140:
136:
132:
128:
124:
120:
116:
112:
108:
104:
100:
97:
94:are those of
93:
77:
74:
71:
64:
48:
44:
40:
32:
28:
24:
16:
5396:
5366:
5344:
5319:
5297:
5276:
5264:
5252:
5240:
5228:
5216:
5204:
5192:
5159:Field theory
5124:
5073:
5051:
5047:
5043:
5013:
5001:Galois group
4996:
4992:
4988:
4984:
4980:
4976:
4972:
4968:
4923:
4893:
4886:
4877:
4869:
4865:
4831:
4798:
4794:
4790:
4786:
4782:
4778:
4774:
4736:
4716:
4046:
4041:
4037:
4033:
4029:
4025:
4021:
4017:
4013:
4009:
4005:
4003:
3998:
3994:
3990:
3986:
3982:
3978:
3974:
3970:
3966:
3962:
3930:
3922:
3890:
3886:
3881:is called a
3878:
3874:
3870:
3866:
3865:, such that
3862:
3858:
3854:
3850:
3842:
3838:
3834:
3826:
3822:
3792:
3662:
3658:
3654:
3646:
3645:Every field
3644:
3637:
3493:
3489:
3458:
3456:
3451:
3447:
3442:
3438:
3431:
3425:
3421:
3417:
3411:
3407:
3403:
3397:
3393:
3389:
3385:
3379:
3375:
3371:
3367:
3360:
3356:
3352:
3348:
3344:
3338:
3334:
3330:
3325:
3321:
3317:
3313:
3305:
3301:
3299:
3294:
3223:
3219:
3217:vector space
3212:
3207:
3203:
3199:
3194:
3190:
3186:
3184:
3179:
3171:
3163:
3155:
3151:
3147:
3086:
3074:
3042:
3040:
3020:
3016:
3012:
3008:
3001:
2997:
2990:
2926:
2918:
2913:
2908:
2904:
2896:
2892:
2888:
2880:
2876:
2872:
2868:
2860:
2856:
2852:
2850:
2845:
2770:
2707:finite field
2702:
2699:prime number
2694:
2692:
2687:
2683:
2679:
2673:
2668:
2660:
2656:
2582:
2541:
2537:
2533:
2529:
2525:
2517:
2506:
2504:
2499:
2435:
2122:
1849:
1781:
1688:
1390:real numbers
1365:
1357:
1340:
1330:
1325:
1321:
1309:
1305:
1303:
1293:
1289:
1281:
1277:
1272:
1268:
1264:
1261:
1250:
1245:
1241:
1233:
1227:
1223:
1220:
1214:is called a
1211:
1206:is called a
1202:
1198:
1194:
1189:
1185:
1181:
1171:
1167:
986:
982:
978:
970:
966:
962:
958:
954:
950:
946:
939:
936:
935:) (read as "
932:
928:
924:
920:
916:
912:
908:
904:
900:
896:
866:
649:
644:
640:
636:
628:
626:
587:
579:vector space
574:
570:
540:
508:subextension
507:
503:
499:
495:
491:
487:
483:
481:
476:
472:
440:
436:
432:
428:
424:
420:
416:
414:
402:
371:real numbers
364:
317:
173:
171:
147:
143:real numbers
126:
122:
118:
114:
110:
106:
102:
95:
91:
30:
20:
15:
5341:Lang, Serge
5322:, Waltham:
5221:McCoy (1968
5007:are called
3993:where both
3821:, a subset
3185:An element
3176:irreducible
3077:if it is a
3041:An element
2989:defined on
2925:defined on
2885:polynomials
2775:prime field
2579:irreducible
1689:The field
1102:instead of
969:, and that
23:mathematics
5369:, Boston:
5290:References
5109:local ring
4860:is called
4801:such that
4765:is called
4032:such that
3961:such that
3829:is called
3563:, because
3083:polynomial
2848:elements.
1850:The field
1236:adjunction
471:(read as "
431:or simply
375:isomorphic
99:restricted
5403:EMS Press
5060:subgroups
5056:bijection
4874:separable
4862:separable
4769:if every
4694:and thus
4289:in which
4274:⟩
4261:−
4248:⟨
4184:−
3273:−
3259:…
3131:−
2899:) is the
2863:) of all
2787:
2404:∈
2380:∣
2091:∈
2067:∣
1982:∈
1970:∣
1754:∈
1742:∣
1349:morphisms
1334:injective
1328:is used.
1132:…
1074:…
1019:…
959:generated
834:⋅
583:dimension
433:extension
320:1 – 1 = 0
287:, and is
232:⊆
75:⊆
33:(denoted
5414:Category
5379:68015225
5343:(2004),
5153:See also
5084:subrings
4991:for all
4625:one has
4329:are the
2914:Given a
1531:because
1362:Examples
174:subfield
168:Subfield
156:through
131:addition
123:subfield
5405:, 2001
5345:Algebra
5005:abelian
2697:is any
2591:maximal
2513:of the
1351:in the
1300:Caveats
989:. When
985:) over
494:, then
423:, then
293:inverse
27:algebra
5377:
5355:
5330:
5308:
4767:normal
4097:where
3450:and 1/
3293:where
3158:, the
3004:, the
2659:which
2593:, and
1413:, and
1345:ideals
942:adjoin
588:degree
581:. The
427:is an
289:closed
220:subset
137:, the
109:is an
63:fields
5185:Notes
4979:with
4868:over
4162:is a
3833:over
3651:up to
3178:over
2887:over
2844:with
2769:with
2587:ideal
2540:with
2509:as a
1682:(the
1341:Every
1292:over
1184:. If
1180:over
1174:) is
973:is a
965:over
573:is a
510:) of
475:over
218:is a
197:field
195:of a
121:is a
5375:LCCN
5353:ISBN
5328:ISBN
5306:ISBN
5080:ring
4987:) =
4880:. A
4658:and
4353:and
4309:and
4164:root
4142:and
3997:and
3587:and
3429:(if
3415:and
3342:and
3324:and
3079:root
3035:and
2701:and
2661:does
2522:root
2158:and
1271:) /
1210:and
1201:) /
923:and
911:and
762:and
678:and
639:and
502:(or
479:").
403:The
342:and
133:and
117:and
29:, a
4995:in
4934:Aut
4872:is
4773:in
4721:of
4333:of
4044:).
4004:If
3933:of
3885:of
3849:of
3825:of
3779:).
3457:An
3434:≠ 0
3424:) /
3420:(1/
3410:) /
3396:) /
3378:) /
3355:) /
3337:) /
3316:in
3312:of
3206:) /
3189:of
3170:of
3150:of
2693:If
2686:of
2671:).
2667:of
2581:in
2577:is
1316:or
1288:in
1251:In
1244:to
1240:of
977:of
961:by
899:of
624:.
506:or
482:If
435:of
415:If
318:As
125:of
113:of
101:to
21:In
5416::
5401:,
5395:,
5373:,
5347:,
5326:,
5304:,
5149:.
5111:.
5066:.
5011:.
4975:→
4971::
4200:0.
4036:=
4020:=
3965:=
3448:st
3446:,
3441:±
3408:st
3401:,
3392:±
3374:,
3351:)(
3182:.
3134:2.
3023:.
2784:GF
2502:.
1355:.
1308:/
1296:.
1248:.
538:.
400:.
315:.
172:A
164:.
5052:K
5048:L
5044:F
5030:K
5026:/
5022:L
4997:K
4993:x
4989:x
4985:x
4983:(
4981:α
4977:L
4973:L
4969:α
4952:)
4949:K
4945:/
4941:L
4938:(
4910:K
4906:/
4902:L
4878:K
4870:K
4866:L
4848:K
4844:/
4840:L
4817:K
4813:/
4809:L
4799:F
4795:L
4791:K
4789:/
4787:F
4783:L
4779:L
4775:K
4753:K
4749:/
4745:L
4702:t
4682:,
4677:3
4673:t
4669:=
4666:y
4644:2
4640:t
4636:=
4633:x
4613:,
4610:x
4606:/
4602:y
4599:=
4596:t
4576:}
4573:y
4570:{
4550:)
4547:x
4544:(
4540:Q
4535:/
4531:)
4528:y
4525:,
4522:x
4519:(
4515:Q
4494:}
4491:x
4488:{
4468:)
4465:x
4462:(
4458:Q
4453:/
4449:)
4446:y
4443:,
4440:x
4437:(
4433:Q
4411:Q
4390:}
4387:x
4384:{
4364:.
4361:Y
4341:X
4317:y
4297:x
4277:,
4269:3
4265:X
4256:2
4252:Y
4244:/
4240:]
4237:Y
4234:[
4231:)
4228:X
4225:(
4221:Q
4197:=
4192:3
4188:x
4179:2
4175:y
4150:y
4130:,
4126:Q
4105:x
4085:,
4081:Q
4076:/
4072:)
4069:y
4066:,
4063:x
4060:(
4056:Q
4042:S
4040:(
4038:K
4034:L
4030:S
4026:S
4024:(
4022:K
4018:L
4014:S
4010:K
4008:/
4006:L
3999:K
3995:L
3991:K
3989:/
3987:L
3983:K
3979:K
3975:L
3971:S
3969:(
3967:K
3963:L
3949:K
3945:/
3941:L
3931:S
3909:K
3905:/
3901:L
3891:K
3889:/
3887:L
3879:S
3875:S
3873:(
3871:K
3869:/
3867:L
3863:K
3859:S
3855:K
3853:/
3851:L
3843:S
3839:K
3835:K
3827:L
3823:S
3809:K
3805:/
3801:L
3766:Q
3755:π
3740:Q
3718:Q
3696:R
3674:C
3663:K
3659:K
3655:K
3647:K
3624:.
3620:Q
3597:3
3573:2
3550:Q
3529:)
3524:3
3519:,
3514:2
3509:(
3505:Q
3494:K
3490:L
3476:K
3472:/
3468:L
3452:s
3443:t
3439:s
3432:s
3426:K
3422:s
3418:K
3412:K
3406:(
3404:K
3398:K
3394:t
3390:s
3388:(
3386:K
3380:K
3376:t
3372:s
3370:(
3368:K
3363:)
3361:s
3359:(
3357:K
3353:t
3349:s
3347:(
3345:K
3339:K
3335:s
3333:(
3331:K
3326:t
3322:s
3318:L
3314:K
3306:K
3302:L
3295:d
3281:,
3276:1
3270:d
3266:s
3262:,
3256:,
3251:2
3247:s
3243:,
3240:s
3237:,
3234:1
3224:s
3222:(
3220:K
3215:-
3213:K
3208:K
3204:s
3202:(
3200:K
3195:K
3191:L
3187:s
3180:K
3172:x
3164:x
3156:K
3152:L
3148:x
3126:2
3122:x
3099:2
3087:K
3075:K
3061:K
3057:/
3053:L
3043:x
3021:K
3017:V
3013:V
3011:(
3009:K
3002:K
2998:V
2991:M
2972:C
2951:.
2948:)
2945:M
2942:(
2938:C
2927:M
2919:M
2909:K
2905:K
2897:X
2895:(
2893:K
2889:K
2881:X
2879:(
2877:K
2873:K
2869:X
2861:X
2859:(
2857:K
2853:K
2846:p
2831:Z
2827:p
2823:/
2818:Z
2814:=
2809:p
2804:F
2799:=
2796:)
2793:p
2790:(
2771:p
2753:n
2749:p
2743:F
2738:=
2735:)
2730:n
2726:p
2722:(
2719:F
2716:G
2703:n
2695:p
2688:K
2684:L
2680:K
2669:X
2657:K
2643:)
2640:1
2637:+
2632:2
2628:X
2624:(
2620:/
2616:]
2613:X
2610:[
2607:K
2604:=
2601:L
2583:K
2565:1
2562:+
2557:2
2553:X
2542:x
2538:x
2534:K
2530:X
2528:(
2526:f
2518:K
2507:K
2500:p
2484:p
2479:Q
2445:Q
2417:.
2413:}
2408:Q
2401:d
2398:,
2395:c
2392:,
2389:b
2386:,
2383:a
2375:3
2371:)
2365:3
2360:+
2355:2
2350:(
2347:d
2344:+
2339:2
2335:)
2329:3
2324:+
2319:2
2314:(
2311:c
2308:+
2305:)
2300:3
2295:+
2290:2
2285:(
2282:b
2279:+
2276:a
2272:{
2268:=
2258:)
2253:3
2248:+
2243:2
2238:(
2234:Q
2230:=
2223:)
2218:3
2213:,
2208:2
2203:(
2199:Q
2171:,
2167:Q
2146:)
2141:2
2136:(
2132:Q
2104:,
2100:}
2095:Q
2088:d
2085:,
2082:c
2079:,
2076:b
2073:,
2070:a
2062:6
2057:d
2054:+
2049:3
2044:c
2041:+
2036:2
2031:b
2028:+
2025:a
2021:{
2017:=
2006:}
2001:)
1996:2
1991:(
1986:Q
1979:b
1976:,
1973:a
1965:3
1960:b
1957:+
1954:a
1950:{
1946:=
1935:)
1930:3
1925:(
1920:)
1915:2
1910:(
1905:Q
1901:=
1893:)
1887:3
1882:,
1877:2
1871:(
1866:Q
1834:}
1828:2
1823:,
1820:1
1816:{
1795:,
1791:Q
1767:,
1763:}
1758:Q
1751:b
1748:,
1745:a
1737:2
1732:b
1729:+
1726:a
1722:{
1718:=
1715:)
1710:2
1705:(
1701:Q
1668:c
1663:=
1660:]
1656:Q
1652::
1648:R
1644:[
1625:.
1622:)
1619:i
1616:(
1612:R
1608:=
1604:C
1582:R
1577:/
1572:C
1551:}
1548:i
1545:,
1542:1
1539:{
1519:2
1516:=
1513:]
1509:R
1505::
1501:C
1497:[
1476:Q
1471:/
1466:C
1444:Q
1422:R
1400:R
1375:C
1326:K
1324::
1322:L
1310:K
1306:L
1294:K
1290:s
1282:s
1280:(
1278:K
1273:K
1269:s
1267:(
1265:K
1246:K
1242:S
1230:)
1228:S
1226:(
1224:K
1212:s
1203:K
1199:s
1197:(
1195:K
1190:s
1186:S
1182:K
1172:S
1170:(
1168:K
1154:,
1151:)
1148:}
1143:n
1139:x
1135:,
1129:,
1124:1
1120:x
1116:{
1113:(
1110:K
1090:)
1085:n
1081:x
1077:,
1071:,
1066:1
1062:x
1058:(
1055:K
1035:}
1030:n
1026:x
1022:,
1016:,
1011:1
1007:x
1003:{
1000:=
997:S
987:K
983:S
981:(
979:K
971:S
967:K
963:S
955:S
953:(
951:K
947:S
937:K
933:S
931:(
929:K
925:S
921:K
917:L
913:S
909:K
905:L
901:L
897:S
883:K
879:/
875:L
852:.
849:]
846:K
843::
840:L
837:[
831:]
828:L
825::
822:M
819:[
816:=
813:]
810:K
807::
804:M
801:[
778:L
774:/
770:M
750:K
746:/
742:L
722:K
718:/
714:M
694:L
690:/
686:M
666:K
662:/
658:L
612:]
609:K
606::
603:L
600:[
577:-
575:K
571:L
557:K
553:/
549:L
526:K
522:/
518:L
496:F
492:K
488:F
484:L
477:K
473:L
459:K
455:/
451:L
437:K
425:L
421:L
417:K
388:0
350:L
330:K
303:K
275:1
255:L
235:L
229:K
206:L
183:K
127:L
119:K
115:K
107:L
103:K
96:L
92:K
78:L
72:K
49:K
45:/
41:L
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.