2129:
2442:
1867:
2200:
2124:{\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}}}
2437:{\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}}}
5156:. 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
1788:
1342:
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
2853:
5109:
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
4298:
4039:). 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
3550:
2205:
1872:
4571:
4489:
4106:
2778:
1646:
1856:
1175:
3302:
2167:
1604:
1498:
1056:
1111:
4973:
2972:
2507:
1691:
4221:
4151:
3645:
2664:
2192:
1816:
4433:
3788:
3762:
3740:
3718:
3696:
3572:
2994:
2467:
1466:
1444:
1422:
1397:
256:
99:
3620:
3596:
3122:
4703:
3155:
873:
4667:
2586:
1706:
1572:
4634:
3653:
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.
4597:
4515:
4411:
5051:
4931:
4869:
4838:
4774:
3970:
3930:
3830:
3497:
3082:
904:
799:
771:
743:
715:
687:
578:
547:
480:
70:
4385:
1540:
4723:
4362:
4338:
4318:
4171:
4126:
409:
371:
351:
324:
296:
276:
227:
204:
633:
2790:
4729:
3016:
5367:
5065:). The significance of Galois extensions and Galois groups is that they allow a complete description of the intermediate fields: there is a
5074:
4599:
is a transcendence basis that does not generates the whole extension. However the extension is purely transcendental since, if one set
2477:. Another extension field of the rationals, which is also important in number theory, although not a finite extension, is the field of
5342:
3996:, but, however, there are extensions with this property which are not purely transcendental—a class of such extensions take the form
5320:
3178:
4226:
1226:
3222:
is a finite extension. In this case the degree of the extension equals the degree of the minimal polynomial, and a basis of the
3510:
5413:
5359:
4740:
of a rational variety is equivalent with the problem of finding a transcendence basis that generates the whole extension.
1694:
4520:
4438:
5408:
5140:" on associated algebraic objects. For example, given a real vector space, one can produce a complex vector space via
3904:. All transcendence bases have the same cardinality, equal to the transcendence degree of the extension. An extension
597:
4061:
2722:
5169:
1609:
5334:
1821:
5430:
4899:
3841:
1267:
1116:
109:
3240:
5174:
4737:
3799:
2137:
1577:
1471:
1263:
1003:
593:
415:
389:
1369:
Henceforth, we will suppress the injective homomorphism and assume that we are dealing with actual subfields.
3507:. Equivalently, an algebraic extension is an extension that is generated by algebraic elements. For example,
384:, which is itself a subfield of the complex numbers. More generally, the field of rational numbers is (or is
1331:
or any other kind of division. Instead the slash expresses the word "over". In some literature the notation
1061:
160:
5403:
5098:
4781:
3186:
2589:
2470:
638:
The degree of an extension is 1 if and only if the two fields are equal. In this case, the extension is a
303:
299:
4939:
1354:
non-zero ring homomorphism between fields is injective because fields do not possess nontrivial proper
2943:
2483:
1650:
5157:
5153:
5137:
5131:
5106:
4884:
3857:
2933:
4180:
5145:
4872:
4131:
3893:
3625:
3043:
2607:
2597:
2172:
1796:
1355:
207:
73:
4416:
3771:
3745:
3723:
3701:
3679:
3555:
2977:
2450:
1449:
1427:
1405:
1380:
235:
78:
5090:
4733:
3601:
3577:
3103:
3089:
2911:
2675:
2532:
1363:
1344:
172:
5362:, vol. 211 (Corrected fourth printing, revised third ed.), New York: Springer-Verlag,
1783:{\displaystyle \mathbb {Q} ({\sqrt {2}})=\left\{a+b{\sqrt {2}}\mid a,b\in \mathbb {Q} \right\},}
5385:
5363:
5338:
5316:
5189:
5184:
5111:
5019:
4672:
4341:
3320:
3127:
3047:
3005:
2997:
2875:
1347:
1296:
807:
4639:
2558:
1545:
5141:
4892:
4777:
4602:
3170:
1218:
164:
4576:
4494:
4390:
5381:
5179:
5115:
5086:
4174:
2926:
2686:
2525:
377:
5028:
4908:
4846:
4815:
4751:
3947:
3907:
3807:
3474:
3059:
881:
776:
748:
720:
692:
664:
555:
524:
457:
47:
4367:
3672:, and also the smallest extension field such that every polynomial with coefficients in
1503:
5312:
5102:
4902:
states that every finite separable extension has a primitive element (i.e. is simple).
4708:
4347:
4323:
4303:
4156:
4111:
3650:
1328:
985:
394:
356:
336:
309:
281:
261:
212:
189:
149:
145:
606:
5424:
5015:
2601:
2521:
2478:
2474:
2194:
of degree 2 and 4 respectively. It is also a simple extension, as one can show that
1324:
168:
17:
5011:
4976:
3227:
2717:
2709:
589:
2848:{\displaystyle \operatorname {GF} (p)=\mathbb {F} _{p}=\mathbb {Z} /p\mathbb {Z} }
2785:
1400:
385:
381:
153:
33:
29:
Construction of a larger algebraic field by "adding elements" to a smaller field
5351:
5149:
5119:
4027:
is a transcendence basis of the extension, it doesn't necessarily follow that
3093:
2895:
302:
under the operations of addition, subtraction, multiplication, and taking the
5066:
3856:. The largest cardinality of an algebraically independent set is called the
4728:
Purely transcendental extensions of an algebraically closed field occur as
5144:. In addition to vector spaces, one can perform extension of scalars for
5070:
1359:
141:
5094:
2701:
in which the given polynomial splits into a product of linear factors.
37:
5010:. When the extension is Galois this automorphism group is called the
418:
of a subfield is the same as the characteristic of the larger field.
230:
5114:
to the reals or the quaternions. CSAs can be further generalized to
258:
that is a field with respect to the field operations inherited from
5105:(no non-trivial 2-sided ideals, just as for a field) and where the
3661:
1270:, which does not hold true for fields of non-zero characteristic.
5389:
3124:
is algebraic over the rational numbers, because it is a root of
4517:
is a transcendence basis that does not generates the extension
1446:
in turn is an extension field of the field of rational numbers
2516:
It is common to construct an extension field of a given field
2716:
is a positive integer, there is a unique (up to isomorphism)
1266:
0, every finite extension is a simple extension. This is the
4293:{\displaystyle \mathbb {Q} (X)/\langle Y^{2}-X^{3}\rangle ,}
2918:. This field of rational functions is an extension field of
4887:, i.e., has no repeated roots in an algebraic closure over
3984:). Such an extension has the property that all elements of
3848:
if no non-trivial polynomial relation with coefficients in
2996:
if we identify every complex number with the corresponding
156:; the real numbers are a subfield of the complex numbers.
2685:
By iterating the above construction, one can construct a
1818:
also clearly a simple extension. The degree is 2 because
4895:
is a field extension that is both normal and separable.
3331:. This results from the preceding characterization: if
1323:
is purely formal and does not imply the formation of a
5053:, one is often interested in the intermediate fields
5031:
4942:
4911:
4849:
4818:
4754:
4711:
4675:
4642:
4605:
4579:
4523:
4497:
4441:
4419:
4393:
4370:
4350:
4326:
4306:
4229:
4183:
4159:
4134:
4114:
4064:
3950:
3910:
3810:
3774:
3748:
3726:
3704:
3682:
3628:
3604:
3580:
3558:
3545:{\displaystyle \mathbb {Q} ({\sqrt {2}},{\sqrt {3}})}
3513:
3477:
3243:
3130:
3106:
3062:
2980:
2946:
2793:
2725:
2610:
2561:
2486:
2453:
2203:
2175:
2140:
1870:
1824:
1799:
1709:
1653:
1612:
1580:
1548:
1506:
1474:
1452:
1430:
1408:
1383:
1119:
1064:
1006:
884:
810:
779:
751:
723:
695:
667:
609:
558:
527:
460:
397:
359:
339:
312:
284:
278:. Equivalently, a subfield is a subset that contains
264:
238:
215:
192:
81:
50:
5118:, where the base field is replaced by a commutative
5014:of the extension. Extensions whose Galois group is
4840:is normal and which is minimal with this property.
3026:), consisting of the rational functions defined on
5045:
4967:
4925:
4863:
4832:
4768:
4717:
4697:
4661:
4628:
4591:
4565:
4509:
4483:
4427:
4405:
4379:
4356:
4332:
4312:
4292:
4215:
4165:
4145:
4120:
4100:
3964:
3940:if and only if there exists a transcendence basis
3924:
3824:
3782:
3756:
3734:
3712:
3690:
3639:
3614:
3590:
3566:
3544:
3491:
3296:
3149:
3116:
3076:
2988:
2966:
2847:
2772:
2674:contain an element whose square is −1 (namely the
2658:
2580:
2501:
2461:
2436:
2186:
2161:
2123:
1850:
1810:
1782:
1685:
1640:
1598:
1566:
1534:
1492:
1460:
1438:
1416:
1391:
1169:
1105:
1050:
898:
867:
793:
765:
737:
709:
681:
627:
572:
541:
474:
403:
365:
345:
318:
290:
270:
250:
221:
198:
93:
64:
5101:(CSAs) – ring extensions over a field, which are
4566:{\displaystyle \mathbb {Q} (x,y)/\mathbb {Q} (x)}
4484:{\displaystyle \mathbb {Q} (x,y)/\mathbb {Q} (x)}
5378:Introduction To Modern Algebra, Revised Edition
5148:defined over the field, such as polynomials or
4101:{\displaystyle \mathbb {Q} (x,y)/\mathbb {Q} ,}
4875:if the minimal polynomial of every element of
3664:an isomorphism the largest extension field of
3394:is also finite, as well as the sub extensions
2773:{\displaystyle GF(p^{n})=\mathbb {F} _{p^{n}}}
1606:is finite. This is a simple extension because
1641:{\displaystyle \mathbb {C} =\mathbb {R} (i).}
926:. It is the intersection of all subfields of
8:
4792:completely factors into linear factors over
4586:
4580:
4504:
4498:
4400:
4394:
4284:
4258:
2784:elements; this is an extension field of the
1851:{\displaystyle \left\{1,{\sqrt {2}}\right\}}
1561:
1549:
1158:
1126:
1045:
1013:
3499:is an extension such that every element of
1170:{\displaystyle K(\{x_{1},\ldots ,x_{n}\}),}
3297:{\displaystyle 1,s,s^{2},\ldots ,s^{d-1},}
2974:It is a transcendental extension field of
658:is an extension that has a finite degree.
646:. Extensions of degree 2 and 3 are called
140:. For example, under the usual notions of
5035:
5030:
4954:
4943:
4941:
4915:
4910:
4853:
4848:
4822:
4817:
4758:
4753:
4710:
4686:
4674:
4653:
4641:
4615:
4604:
4578:
4550:
4549:
4544:
4525:
4524:
4522:
4496:
4468:
4467:
4462:
4443:
4442:
4440:
4421:
4420:
4418:
4392:
4369:
4349:
4325:
4305:
4278:
4265:
4253:
4231:
4230:
4228:
4201:
4188:
4182:
4158:
4136:
4135:
4133:
4113:
4091:
4090:
4085:
4066:
4065:
4063:
3954:
3949:
3914:
3909:
3814:
3809:
3776:
3775:
3773:
3750:
3749:
3747:
3728:
3727:
3725:
3706:
3705:
3703:
3684:
3683:
3681:
3630:
3629:
3627:
3605:
3603:
3581:
3579:
3560:
3559:
3557:
3532:
3522:
3515:
3514:
3512:
3481:
3476:
3319:form a subextension, which is called the
3308:is the degree of the minimal polynomial.
3279:
3260:
3242:
3135:
3129:
3107:
3105:
3066:
3061:
2982:
2981:
2979:
2948:
2947:
2945:
2841:
2840:
2832:
2828:
2827:
2818:
2814:
2813:
2792:
2762:
2757:
2753:
2752:
2739:
2724:
2641:
2629:
2609:
2566:
2560:
2493:
2489:
2488:
2485:
2455:
2454:
2452:
2418:
2417:
2384:
2373:
2363:
2348:
2337:
2327:
2308:
2298:
2261:
2251:
2244:
2243:
2226:
2216:
2209:
2208:
2204:
2202:
2177:
2176:
2174:
2162:{\displaystyle \mathbb {Q} ({\sqrt {2}})}
2149:
2142:
2141:
2139:
2105:
2104:
2070:
2057:
2044:
2004:
1996:
1995:
1973:
1938:
1923:
1915:
1914:
1895:
1885:
1876:
1875:
1871:
1869:
1836:
1823:
1801:
1800:
1798:
1768:
1767:
1745:
1718:
1711:
1710:
1708:
1677:
1676:
1666:
1665:
1658:
1657:
1652:
1622:
1621:
1614:
1613:
1611:
1599:{\displaystyle \mathbb {C} /\mathbb {R} }
1592:
1591:
1586:
1582:
1581:
1579:
1547:
1519:
1518:
1511:
1510:
1505:
1493:{\displaystyle \mathbb {C} /\mathbb {Q} }
1486:
1485:
1480:
1476:
1475:
1473:
1454:
1453:
1451:
1432:
1431:
1429:
1410:
1409:
1407:
1385:
1384:
1382:
1152:
1133:
1118:
1094:
1075:
1063:
1051:{\displaystyle S=\{x_{1},\ldots ,x_{n}\}}
1039:
1020:
1005:
888:
883:
809:
783:
778:
755:
750:
727:
722:
699:
694:
671:
666:
608:
562:
557:
531:
526:
464:
459:
396:
358:
338:
311:
283:
263:
237:
214:
191:
80:
54:
49:
5291:
5279:
5267:
5255:
5243:
5219:
5207:
5069:between the intermediate fields and the
1358:, so field extensions are precisely the
5200:
5085:Field extensions can be generalized to
4744:Normal, separable and Galois extensions
5097:. A closer non-commutative analog are
5073:of the Galois group, described by the
3868:. It is always possible to find a set
1399:is an extension field of the field of
1106:{\displaystyle K(x_{1},\ldots ,x_{n})}
5231:
7:
5075:fundamental theorem of Galois theory
4223:Such an extension can be defined as
4058:For example, consider the extension
3208:if and only if the simple extension
454:. Such a field extension is denoted
159:Field extensions are fundamental in
3660:has an algebraic closure, which is
1678:
1500:is also a field extension. We have
596:of this vector space is called the
501:, which is in turn an extension of
5309:A First Course In Abstract Algebra
5158:extension of scalars: applications
5136:Given a field extension, one can "
3720:, but not an algebraic closure of
1697:), so this extension is infinite.
914:, there is a smallest subfield of
801:are finite. In this case, one has
25:
4968:{\displaystyle {\text{Aut}}(L/K)}
4808:, which is an extension field of
3872:, algebraically independent over
1243:is often said to result from the
4725:generates the whole extension.
3649:A simple extension is algebraic
2967:{\displaystyle \mathbb {C} (M).}
2600:generated by this polynomial is
2502:{\displaystyle \mathbb {Q} _{p}}
1686:{\displaystyle ={\mathfrak {c}}}
1295:) is isomorphic to the field of
333:, the latter definition implies
3676:has a root in it. For example,
1232:An extension field of the form
450:, and this pair of fields is a
388:to) a subfield of any field of
4962:
4948:
4560:
4554:
4541:
4529:
4478:
4472:
4459:
4447:
4250:
4244:
4241:
4235:
4216:{\displaystyle y^{2}-x^{3}=0.}
4082:
4070:
3742:, as it is not algebraic over
3539:
3519:
3339:are algebraic, the extensions
2958:
2952:
2922:. This extension is infinite.
2806:
2800:
2745:
2732:
2653:
2634:
2626:
2620:
2381:
2360:
2345:
2324:
2315:
2295:
2268:
2248:
2233:
2213:
2156:
2146:
2134:is an extension field of both
1725:
1715:
1670:
1654:
1632:
1626:
1523:
1507:
1161:
1123:
1100:
1068:
859:
847:
841:
829:
823:
811:
745:is finite if and only if both
622:
610:
152:are an extension field of the
101:, such that the operations of
1:
5360:Graduate Texts in Mathematics
4387:Obviously, the singleton set
4146:{\displaystyle \mathbb {Q} ,}
4023:is purely transcendental and
3852:exists among the elements of
3640:{\displaystyle \mathbb {Q} .}
3552:is an algebraic extension of
3185:. This minimal polynomial is
2693:. This is an extension field
2659:{\displaystyle L=K/(X^{2}+1)}
2547:does not contain any element
2543:). Suppose for instance that
2187:{\displaystyle \mathbb {Q} ,}
1811:{\displaystyle \mathbb {Q} ,}
1574:is a basis, so the extension
1377:The field of complex numbers
1199:consists of a single element
5335:Blaisdell Publishing Company
5025:For a given field extension
4796:. Every algebraic extension
4428:{\displaystyle \mathbb {Q} }
3783:{\displaystyle \mathbb {Q} }
3757:{\displaystyle \mathbb {Q} }
3735:{\displaystyle \mathbb {Q} }
3713:{\displaystyle \mathbb {R} }
3691:{\displaystyle \mathbb {C} }
3567:{\displaystyle \mathbb {Q} }
2989:{\displaystyle \mathbb {C} }
2866:, we can consider the field
2462:{\displaystyle \mathbb {Q} }
1695:cardinality of the continuum
1461:{\displaystyle \mathbb {Q} }
1439:{\displaystyle \mathbb {R} }
1417:{\displaystyle \mathbb {R} }
1392:{\displaystyle \mathbb {C} }
373:have the same zero element.
251:{\displaystyle K\subseteq L}
94:{\displaystyle K\subseteq L}
5409:Encyclopedia of Mathematics
4736:. The problem of finding a
4012:are algebraically closed.
3888:) is algebraic. Such a set
3698:is an algebraic closure of
3615:{\displaystyle {\sqrt {3}}}
3591:{\displaystyle {\sqrt {2}}}
3311:The set of the elements of
3117:{\displaystyle {\sqrt {2}}}
3030:, is an extension field of
3004:. More generally, given an
5447:
5307:Fraleigh, John B. (1976),
5129:
4975:, consisting of all field
4905:Given any field extension
3797:
3173:of lowest degree that has
3041:
2555:= −1. Then the polynomial
376:For example, the field of
5311:(2nd ed.), Reading:
4900:primitive element theorem
3842:algebraically independent
2666:is an extension field of
1793:is an extension field of
1287:is not finite, the field
1268:primitive element theorem
171:, and are widely used in
5329:Herstein, I. N. (1964),
5175:Glossary of field theory
4804:admits a normal closure
4738:rational parametrization
4698:{\displaystyle y=t^{3},}
3992:are transcendental over
3804:Given a field extension
3800:Transcendental extension
3794:Transcendental extension
3668:which is algebraic over
3315:that are algebraic over
3177:as a root is called the
3150:{\displaystyle x^{2}-2.}
878:Given a field extension
868:{\displaystyle =\cdot .}
552:Given a field extension
306:of a nonzero element of
5376:McCoy, Neal H. (1968),
5099:central simple algebras
4843:An algebraic extension
4748:An algebraic extension
4662:{\displaystyle x=t^{2}}
4413:is transcendental over
4128:is transcendental over
2940:is a field, denoted by
2914:of the polynomial ring
2894:) are fractions of two
2689:of any polynomial from
2581:{\displaystyle X^{2}+1}
2535:for a given polynomial
2531:in order to "create" a
2471:algebraic number fields
1858:can serve as a basis.
1567:{\displaystyle \{1,i\}}
161:algebraic number theory
5404:"Extension of a field"
5047:
4969:
4933:, we can consider its
4927:
4865:
4834:
4782:irreducible polynomial
4770:
4719:
4699:
4663:
4630:
4629:{\displaystyle t=y/x,}
4593:
4567:
4511:
4485:
4429:
4407:
4381:
4358:
4334:
4314:
4294:
4217:
4167:
4147:
4122:
4102:
3966:
3926:
3826:
3784:
3768:is not algebraic over
3758:
3736:
3714:
3692:
3641:
3616:
3592:
3568:
3546:
3493:
3298:
3151:
3118:
3078:
2990:
2968:
2849:
2774:
2660:
2582:
2503:
2463:
2438:
2188:
2163:
2125:
1852:
1812:
1784:
1687:
1642:
1600:
1568:
1536:
1494:
1462:
1440:
1418:
1393:
1273:If a simple extension
1171:
1107:
1058:is finite, one writes
1052:
900:
869:
795:
767:
739:
711:
683:
629:
574:
543:
515:intermediate extension
476:
405:
367:
347:
320:
292:
272:
252:
223:
200:
163:, and in the study of
95:
66:
5154:group representations
5048:
4970:
4928:
4898:A consequence of the
4866:
4835:
4771:
4720:
4700:
4664:
4631:
4594:
4592:{\displaystyle \{y\}}
4568:
4512:
4510:{\displaystyle \{x\}}
4486:
4430:
4408:
4406:{\displaystyle \{x\}}
4382:
4359:
4335:
4315:
4295:
4218:
4168:
4148:
4123:
4103:
3967:
3936:purely transcendental
3927:
3827:
3785:
3759:
3737:
3715:
3693:
3642:
3617:
3593:
3569:
3547:
3494:
3299:
3152:
3119:
3096:with coefficients in
3079:
3056:of a field extension
2991:
2969:
2934:meromorphic functions
2882:with coefficients in
2850:
2775:
2661:
2583:
2504:
2473:and are important in
2464:
2447:Finite extensions of
2439:
2189:
2164:
2126:
1853:
1813:
1785:
1688:
1643:
1601:
1569:
1537:
1495:
1463:
1441:
1419:
1394:
1172:
1108:
1053:
901:
870:
796:
768:
740:
712:
684:
661:Given two extensions
630:
575:
544:
477:
406:
380:is a subfield of the
368:
348:
321:
293:
273:
253:
224:
201:
96:
67:
18:Purely transcendental
5146:associative algebras
5132:Extension of scalars
5126:Extension of scalars
5029:
4940:
4909:
4847:
4816:
4752:
4709:
4673:
4640:
4603:
4577:
4521:
4495:
4491:is algebraic; hence
4439:
4417:
4391:
4368:
4348:
4324:
4304:
4227:
4181:
4157:
4132:
4112:
4062:
3948:
3908:
3858:transcendence degree
3808:
3772:
3746:
3724:
3702:
3680:
3626:
3602:
3578:
3556:
3511:
3475:
3241:
3128:
3104:
3060:
2978:
2944:
2791:
2723:
2608:
2559:
2484:
2451:
2201:
2173:
2138:
1868:
1822:
1797:
1707:
1651:
1610:
1578:
1546:
1504:
1472:
1450:
1428:
1406:
1381:
1350:between two fields.
1117:
1062:
1004:
938:, and is denoted by
882:
808:
777:
749:
721:
693:
665:
648:quadratic extensions
607:
556:
525:
458:
395:
357:
337:
310:
282:
262:
236:
213:
190:
79:
48:
5152:and the associated
5089:which consist of a
5046:{\displaystyle L/K}
4926:{\displaystyle L/K}
4864:{\displaystyle L/K}
4833:{\displaystyle L/K}
4788:that has a root in
4769:{\displaystyle L/K}
4342:equivalence classes
3965:{\displaystyle L/K}
3925:{\displaystyle L/K}
3894:transcendence basis
3825:{\displaystyle L/K}
3622:are algebraic over
3492:{\displaystyle L/K}
3470:algebraic extension
3465:are all algebraic.
3447:). It follows that
3077:{\displaystyle L/K}
3044:Algebraic extension
3038:Algebraic extension
2596:, consequently the
2509:for a prime number
899:{\displaystyle L/K}
794:{\displaystyle M/L}
766:{\displaystyle L/K}
738:{\displaystyle M/K}
710:{\displaystyle M/L}
682:{\displaystyle L/K}
580:, the larger field
573:{\displaystyle L/K}
542:{\displaystyle L/K}
497:is an extension of
475:{\displaystyle L/K}
65:{\displaystyle L/K}
5107:center of the ring
5043:
5020:abelian extensions
4965:
4935:automorphism group
4923:
4861:
4830:
4766:
4734:rational varieties
4715:
4695:
4659:
4626:
4589:
4563:
4507:
4481:
4435:and the extension
4425:
4403:
4380:{\displaystyle Y.}
4377:
4354:
4330:
4310:
4290:
4213:
4163:
4143:
4118:
4098:
3962:
3922:
3822:
3780:
3754:
3732:
3710:
3688:
3637:
3612:
3588:
3564:
3542:
3503:is algebraic over
3489:
3294:
3204:is algebraic over
3179:minimal polynomial
3165:is algebraic over
3147:
3114:
3084:is algebraic over
3074:
2986:
2964:
2912:field of fractions
2886:; the elements of
2876:rational functions
2845:
2770:
2656:
2578:
2499:
2459:
2434:
2432:
2184:
2159:
2121:
2119:
1848:
1808:
1780:
1683:
1638:
1596:
1564:
1535:{\displaystyle =2}
1532:
1490:
1458:
1436:
1414:
1389:
1364:category of fields
1297:rational fractions
1229:of the extension.
1188:finitely generated
1177:and one says that
1167:
1103:
1048:
960:"). One says that
896:
865:
791:
763:
735:
707:
679:
654:, respectively. A
625:
603:and is denoted by
570:
539:
511:intermediate field
472:
401:
363:
343:
316:
288:
268:
248:
219:
196:
173:algebraic geometry
91:
62:
36:, particularly in
5369:978-0-387-95385-4
5331:Topics In Algebra
5190:Regular extension
5185:Primary extension
5112:Brauer equivalent
4946:
4718:{\displaystyle t}
4357:{\displaystyle X}
4333:{\displaystyle y}
4313:{\displaystyle x}
4166:{\displaystyle y}
4121:{\displaystyle x}
3610:
3586:
3537:
3527:
3376:are finite. Thus
3321:algebraic closure
3112:
3048:Algebraic element
3006:algebraic variety
2998:constant function
2932:, the set of all
2378:
2368:
2342:
2332:
2313:
2303:
2266:
2256:
2231:
2221:
2154:
2075:
2062:
2049:
2009:
1978:
1943:
1928:
1900:
1890:
1841:
1750:
1723:
1348:ring homomorphism
1227:primitive element
642:trivial extension
509:is said to be an
430:is a subfield of
404:{\displaystyle 0}
366:{\displaystyle L}
346:{\displaystyle K}
319:{\displaystyle K}
291:{\displaystyle 1}
271:{\displaystyle L}
222:{\displaystyle L}
199:{\displaystyle K}
16:(Redirected from
5438:
5431:Field extensions
5417:
5392:
5372:
5347:
5325:
5295:
5289:
5283:
5277:
5271:
5265:
5259:
5253:
5247:
5241:
5235:
5229:
5223:
5217:
5211:
5205:
5142:complexification
5116:Azumaya algebras
5052:
5050:
5049:
5044:
5039:
4974:
4972:
4971:
4966:
4958:
4947:
4944:
4932:
4930:
4929:
4924:
4919:
4893:Galois extension
4870:
4868:
4867:
4862:
4857:
4839:
4837:
4836:
4831:
4826:
4775:
4773:
4772:
4767:
4762:
4724:
4722:
4721:
4716:
4704:
4702:
4701:
4696:
4691:
4690:
4668:
4666:
4665:
4660:
4658:
4657:
4635:
4633:
4632:
4627:
4619:
4598:
4596:
4595:
4590:
4572:
4570:
4569:
4564:
4553:
4548:
4528:
4516:
4514:
4513:
4508:
4490:
4488:
4487:
4482:
4471:
4466:
4446:
4434:
4432:
4431:
4426:
4424:
4412:
4410:
4409:
4404:
4386:
4384:
4383:
4378:
4363:
4361:
4360:
4355:
4339:
4337:
4336:
4331:
4319:
4317:
4316:
4311:
4299:
4297:
4296:
4291:
4283:
4282:
4270:
4269:
4257:
4234:
4222:
4220:
4219:
4214:
4206:
4205:
4193:
4192:
4177:of the equation
4172:
4170:
4169:
4164:
4152:
4150:
4149:
4144:
4139:
4127:
4125:
4124:
4119:
4107:
4105:
4104:
4099:
4094:
4089:
4069:
3988:except those of
3971:
3969:
3968:
3963:
3958:
3938:
3937:
3931:
3929:
3928:
3923:
3918:
3831:
3829:
3828:
3823:
3818:
3789:
3787:
3786:
3781:
3779:
3767:
3763:
3761:
3760:
3755:
3753:
3741:
3739:
3738:
3733:
3731:
3719:
3717:
3716:
3711:
3709:
3697:
3695:
3694:
3689:
3687:
3646:
3644:
3643:
3638:
3633:
3621:
3619:
3618:
3613:
3611:
3606:
3597:
3595:
3594:
3589:
3587:
3582:
3573:
3571:
3570:
3565:
3563:
3551:
3549:
3548:
3543:
3538:
3533:
3528:
3523:
3518:
3498:
3496:
3495:
3490:
3485:
3456:
3446:
3439:
3425:
3411:
3393:
3375:
3352:
3303:
3301:
3300:
3295:
3290:
3289:
3265:
3264:
3221:
3171:monic polynomial
3156:
3154:
3153:
3148:
3140:
3139:
3123:
3121:
3120:
3115:
3113:
3108:
3083:
3081:
3080:
3075:
3070:
3011:over some field
2995:
2993:
2992:
2987:
2985:
2973:
2971:
2970:
2965:
2951:
2878:in the variable
2854:
2852:
2851:
2846:
2844:
2836:
2831:
2823:
2822:
2817:
2779:
2777:
2776:
2771:
2769:
2768:
2767:
2766:
2756:
2744:
2743:
2665:
2663:
2662:
2657:
2646:
2645:
2633:
2587:
2585:
2584:
2579:
2571:
2570:
2508:
2506:
2505:
2500:
2498:
2497:
2492:
2469:are also called
2468:
2466:
2465:
2460:
2458:
2443:
2441:
2440:
2435:
2433:
2426:
2422:
2421:
2389:
2388:
2379:
2374:
2369:
2364:
2353:
2352:
2343:
2338:
2333:
2328:
2314:
2309:
2304:
2299:
2274:
2267:
2262:
2257:
2252:
2247:
2232:
2227:
2222:
2217:
2212:
2193:
2191:
2190:
2185:
2180:
2168:
2166:
2165:
2160:
2155:
2150:
2145:
2130:
2128:
2127:
2122:
2120:
2113:
2109:
2108:
2076:
2071:
2063:
2058:
2050:
2045:
2023:
2019:
2015:
2014:
2010:
2005:
1999:
1979:
1974:
1952:
1948:
1944:
1939:
1933:
1929:
1924:
1918:
1906:
1902:
1901:
1896:
1891:
1886:
1879:
1857:
1855:
1854:
1849:
1847:
1843:
1842:
1837:
1817:
1815:
1814:
1809:
1804:
1789:
1787:
1786:
1781:
1776:
1772:
1771:
1751:
1746:
1724:
1719:
1714:
1692:
1690:
1689:
1684:
1682:
1681:
1669:
1661:
1647:
1645:
1644:
1639:
1625:
1617:
1605:
1603:
1602:
1597:
1595:
1590:
1585:
1573:
1571:
1570:
1565:
1541:
1539:
1538:
1533:
1522:
1514:
1499:
1497:
1496:
1491:
1489:
1484:
1479:
1468:. Clearly then,
1467:
1465:
1464:
1459:
1457:
1445:
1443:
1442:
1437:
1435:
1423:
1421:
1420:
1415:
1413:
1398:
1396:
1395:
1390:
1388:
1286:
1249:
1248:
1242:
1219:simple extension
1216:
1203:, the extension
1190:
1189:
1176:
1174:
1173:
1168:
1157:
1156:
1138:
1137:
1112:
1110:
1109:
1104:
1099:
1098:
1080:
1079:
1057:
1055:
1054:
1049:
1044:
1043:
1025:
1024:
955:
954:
905:
903:
902:
897:
892:
874:
872:
871:
866:
800:
798:
797:
792:
787:
772:
770:
769:
764:
759:
744:
742:
741:
736:
731:
717:, the extension
716:
714:
713:
708:
703:
688:
686:
685:
680:
675:
656:finite extension
652:cubic extensions
644:
643:
634:
632:
631:
628:{\displaystyle }
626:
601:of the extension
579:
577:
576:
571:
566:
548:
546:
545:
540:
535:
481:
479:
478:
473:
468:
410:
408:
407:
402:
378:rational numbers
372:
370:
369:
364:
352:
350:
349:
344:
332:
325:
323:
322:
317:
297:
295:
294:
289:
277:
275:
274:
269:
257:
255:
254:
249:
228:
226:
225:
220:
205:
203:
202:
197:
165:polynomial roots
116:. In this case,
100:
98:
97:
92:
71:
69:
68:
63:
58:
21:
5446:
5445:
5441:
5440:
5439:
5437:
5436:
5435:
5421:
5420:
5402:
5399:
5382:Allyn and Bacon
5375:
5370:
5350:
5345:
5328:
5323:
5306:
5303:
5298:
5290:
5286:
5278:
5274:
5266:
5262:
5254:
5250:
5242:
5238:
5230:
5226:
5218:
5214:
5206:
5202:
5198:
5180:Tower of fields
5166:
5134:
5128:
5093:and one of its
5087:ring extensions
5083:
5081:Generalizations
5027:
5026:
4938:
4937:
4907:
4906:
4845:
4844:
4814:
4813:
4750:
4749:
4746:
4730:function fields
4707:
4706:
4682:
4671:
4670:
4649:
4638:
4637:
4601:
4600:
4575:
4574:
4519:
4518:
4493:
4492:
4437:
4436:
4415:
4414:
4389:
4388:
4366:
4365:
4346:
4345:
4322:
4321:
4302:
4301:
4274:
4261:
4225:
4224:
4197:
4184:
4179:
4178:
4155:
4154:
4130:
4129:
4110:
4109:
4060:
4059:
3946:
3945:
3935:
3934:
3906:
3905:
3806:
3805:
3802:
3796:
3770:
3769:
3765:
3744:
3743:
3722:
3721:
3700:
3699:
3678:
3677:
3624:
3623:
3600:
3599:
3576:
3575:
3554:
3553:
3509:
3508:
3473:
3472:
3448:
3441:
3427:
3413:
3395:
3377:
3354:
3340:
3275:
3256:
3239:
3238:
3209:
3131:
3126:
3125:
3102:
3101:
3100:. For example,
3058:
3057:
3050:
3042:Main articles:
3040:
2976:
2975:
2942:
2941:
2927:Riemann surface
2812:
2789:
2788:
2758:
2751:
2735:
2721:
2720:
2687:splitting field
2637:
2606:
2605:
2562:
2557:
2556:
2526:polynomial ring
2487:
2482:
2481:
2449:
2448:
2431:
2430:
2380:
2344:
2285:
2281:
2272:
2271:
2236:
2199:
2198:
2171:
2170:
2136:
2135:
2118:
2117:
2034:
2030:
2021:
2020:
2000:
1963:
1959:
1950:
1949:
1934:
1919:
1907:
1884:
1880:
1866:
1865:
1829:
1825:
1820:
1819:
1795:
1794:
1735:
1731:
1705:
1704:
1649:
1648:
1608:
1607:
1576:
1575:
1544:
1543:
1502:
1501:
1470:
1469:
1448:
1447:
1426:
1425:
1404:
1403:
1379:
1378:
1375:
1313:
1274:
1246:
1245:
1233:
1204:
1187:
1186:
1148:
1129:
1115:
1114:
1090:
1071:
1060:
1059:
1035:
1016:
1002:
1001:
968:) is the field
952:
951:
880:
879:
806:
805:
775:
774:
747:
746:
719:
718:
691:
690:
663:
662:
641:
640:
605:
604:
554:
553:
523:
522:
456:
455:
452:field extension
440:extension field
424:
422:Extension field
393:
392:
355:
354:
335:
334:
330:
308:
307:
280:
279:
260:
259:
234:
233:
211:
210:
188:
187:
181:
150:complex numbers
122:extension field
77:
76:
72:) is a pair of
46:
45:
42:field extension
30:
23:
22:
15:
12:
11:
5:
5444:
5442:
5434:
5433:
5423:
5422:
5419:
5418:
5398:
5397:External links
5395:
5394:
5393:
5373:
5368:
5348:
5344:978-1114541016
5343:
5326:
5321:
5313:Addison-Wesley
5302:
5299:
5297:
5296:
5294:, p. 169)
5292:Herstein (1964
5284:
5282:, p. 319)
5280:Fraleigh (1976
5272:
5270:, p. 363)
5268:Fraleigh (1976
5260:
5258:, p. 193)
5256:Herstein (1964
5248:
5246:, p. 298)
5244:Fraleigh (1976
5236:
5234:, p. 116)
5224:
5222:, p. 167)
5220:Herstein (1964
5212:
5210:, p. 293)
5208:Fraleigh (1976
5199:
5197:
5194:
5193:
5192:
5187:
5182:
5177:
5172:
5165:
5162:
5150:group algebras
5138:extend scalars
5130:Main article:
5127:
5124:
5103:simple algebra
5082:
5079:
5057:(subfields of
5042:
5038:
5034:
4964:
4961:
4957:
4953:
4950:
4922:
4918:
4914:
4860:
4856:
4852:
4829:
4825:
4821:
4765:
4761:
4757:
4745:
4742:
4714:
4694:
4689:
4685:
4681:
4678:
4656:
4652:
4648:
4645:
4625:
4622:
4618:
4614:
4611:
4608:
4588:
4585:
4582:
4562:
4559:
4556:
4552:
4547:
4543:
4540:
4537:
4534:
4531:
4527:
4506:
4503:
4500:
4480:
4477:
4474:
4470:
4465:
4461:
4458:
4455:
4452:
4449:
4445:
4423:
4402:
4399:
4396:
4376:
4373:
4353:
4329:
4309:
4289:
4286:
4281:
4277:
4273:
4268:
4264:
4260:
4256:
4252:
4249:
4246:
4243:
4240:
4237:
4233:
4212:
4209:
4204:
4200:
4196:
4191:
4187:
4162:
4142:
4138:
4117:
4097:
4093:
4088:
4084:
4081:
4078:
4075:
4072:
4068:
3961:
3957:
3953:
3932:is said to be
3921:
3917:
3913:
3821:
3817:
3813:
3798:Main article:
3795:
3792:
3778:
3752:
3730:
3708:
3686:
3651:if and only if
3636:
3632:
3609:
3585:
3562:
3541:
3536:
3531:
3526:
3521:
3517:
3488:
3484:
3480:
3293:
3288:
3285:
3282:
3278:
3274:
3271:
3268:
3263:
3259:
3255:
3252:
3249:
3246:
3237:) consists of
3157:If an element
3146:
3143:
3138:
3134:
3111:
3073:
3069:
3065:
3039:
3036:
3017:function field
2984:
2963:
2960:
2957:
2954:
2950:
2862:Given a field
2843:
2839:
2835:
2830:
2826:
2821:
2816:
2811:
2808:
2805:
2802:
2799:
2796:
2765:
2761:
2755:
2750:
2747:
2742:
2738:
2734:
2731:
2728:
2655:
2652:
2649:
2644:
2640:
2636:
2632:
2628:
2625:
2622:
2619:
2616:
2613:
2577:
2574:
2569:
2565:
2496:
2491:
2479:p-adic numbers
2457:
2445:
2444:
2429:
2425:
2420:
2416:
2413:
2410:
2407:
2404:
2401:
2398:
2395:
2392:
2387:
2383:
2377:
2372:
2367:
2362:
2359:
2356:
2351:
2347:
2341:
2336:
2331:
2326:
2323:
2320:
2317:
2312:
2307:
2302:
2297:
2294:
2291:
2288:
2284:
2280:
2277:
2275:
2273:
2270:
2265:
2260:
2255:
2250:
2246:
2242:
2239:
2237:
2235:
2230:
2225:
2220:
2215:
2211:
2207:
2206:
2183:
2179:
2158:
2153:
2148:
2144:
2132:
2131:
2116:
2112:
2107:
2103:
2100:
2097:
2094:
2091:
2088:
2085:
2082:
2079:
2074:
2069:
2066:
2061:
2056:
2053:
2048:
2043:
2040:
2037:
2033:
2029:
2026:
2024:
2022:
2018:
2013:
2008:
2003:
1998:
1994:
1991:
1988:
1985:
1982:
1977:
1972:
1969:
1966:
1962:
1958:
1955:
1953:
1951:
1947:
1942:
1937:
1932:
1927:
1922:
1917:
1913:
1910:
1908:
1905:
1899:
1894:
1889:
1883:
1878:
1874:
1873:
1846:
1840:
1835:
1832:
1828:
1807:
1803:
1791:
1790:
1779:
1775:
1770:
1766:
1763:
1760:
1757:
1754:
1749:
1744:
1741:
1738:
1734:
1730:
1727:
1722:
1717:
1713:
1680:
1675:
1672:
1668:
1664:
1660:
1656:
1637:
1634:
1631:
1628:
1624:
1620:
1616:
1594:
1589:
1584:
1563:
1560:
1557:
1554:
1551:
1531:
1528:
1525:
1521:
1517:
1513:
1509:
1488:
1483:
1478:
1456:
1434:
1412:
1387:
1374:
1371:
1329:quotient group
1312:
1309:
1264:characteristic
1166:
1163:
1160:
1155:
1151:
1147:
1144:
1141:
1136:
1132:
1128:
1125:
1122:
1102:
1097:
1093:
1089:
1086:
1083:
1078:
1074:
1070:
1067:
1047:
1042:
1038:
1034:
1031:
1028:
1023:
1019:
1015:
1012:
1009:
986:generating set
918:that contains
895:
891:
887:
876:
875:
864:
861:
858:
855:
852:
849:
846:
843:
840:
837:
834:
831:
828:
825:
822:
819:
816:
813:
790:
786:
782:
762:
758:
754:
734:
730:
726:
706:
702:
698:
678:
674:
670:
624:
621:
618:
615:
612:
569:
565:
561:
538:
534:
530:
471:
467:
463:
423:
420:
416:characteristic
400:
390:characteristic
362:
342:
315:
287:
267:
247:
244:
241:
218:
195:
180:
177:
146:multiplication
90:
87:
84:
61:
57:
53:
28:
24:
14:
13:
10:
9:
6:
4:
3:
2:
5443:
5432:
5429:
5428:
5426:
5415:
5411:
5410:
5405:
5401:
5400:
5396:
5391:
5387:
5383:
5379:
5374:
5371:
5365:
5361:
5357:
5353:
5349:
5346:
5340:
5336:
5332:
5327:
5324:
5322:0-201-01984-1
5318:
5314:
5310:
5305:
5304:
5300:
5293:
5288:
5285:
5281:
5276:
5273:
5269:
5264:
5261:
5257:
5252:
5249:
5245:
5240:
5237:
5233:
5228:
5225:
5221:
5216:
5213:
5209:
5204:
5201:
5195:
5191:
5188:
5186:
5183:
5181:
5178:
5176:
5173:
5171:
5168:
5167:
5163:
5161:
5159:
5155:
5151:
5147:
5143:
5139:
5133:
5125:
5123:
5121:
5117:
5113:
5108:
5104:
5100:
5096:
5092:
5088:
5080:
5078:
5076:
5072:
5068:
5064:
5061:that contain
5060:
5056:
5040:
5036:
5032:
5023:
5021:
5017:
5013:
5009:
5005:
5001:
4997:
4993:
4989:
4985:
4981:
4978:
4977:automorphisms
4959:
4955:
4951:
4936:
4920:
4916:
4912:
4903:
4901:
4896:
4894:
4890:
4886:
4882:
4878:
4874:
4858:
4854:
4850:
4841:
4827:
4823:
4819:
4811:
4807:
4803:
4799:
4795:
4791:
4787:
4783:
4779:
4763:
4759:
4755:
4743:
4741:
4739:
4735:
4731:
4726:
4712:
4692:
4687:
4683:
4679:
4676:
4654:
4650:
4646:
4643:
4623:
4620:
4616:
4612:
4609:
4606:
4583:
4573:. Similarly,
4557:
4545:
4538:
4535:
4532:
4501:
4475:
4463:
4456:
4453:
4450:
4397:
4374:
4371:
4351:
4343:
4327:
4307:
4287:
4279:
4275:
4271:
4266:
4262:
4254:
4247:
4238:
4210:
4207:
4202:
4198:
4194:
4189:
4185:
4176:
4160:
4140:
4115:
4095:
4086:
4079:
4076:
4073:
4056:
4054:
4050:
4046:
4042:
4038:
4034:
4030:
4026:
4022:
4018:
4013:
4011:
4007:
4003:
3999:
3995:
3991:
3987:
3983:
3979:
3975:
3959:
3955:
3951:
3943:
3939:
3919:
3915:
3911:
3903:
3899:
3895:
3891:
3887:
3883:
3879:
3875:
3871:
3867:
3863:
3859:
3855:
3851:
3847:
3843:
3839:
3835:
3819:
3815:
3811:
3801:
3793:
3791:
3764:(for example
3675:
3671:
3667:
3663:
3659:
3654:
3652:
3647:
3634:
3607:
3583:
3534:
3529:
3524:
3506:
3502:
3486:
3482:
3478:
3471:
3466:
3464:
3460:
3455:
3451:
3444:
3438:
3434:
3430:
3424:
3420:
3416:
3410:
3406:
3402:
3398:
3392:
3388:
3384:
3380:
3373:
3369:
3365:
3361:
3357:
3351:
3347:
3343:
3338:
3334:
3330:
3326:
3322:
3318:
3314:
3309:
3307:
3291:
3286:
3283:
3280:
3276:
3272:
3269:
3266:
3261:
3257:
3253:
3250:
3247:
3244:
3236:
3232:
3229:
3225:
3220:
3216:
3212:
3207:
3203:
3199:
3194:
3192:
3188:
3184:
3180:
3176:
3172:
3168:
3164:
3160:
3144:
3141:
3136:
3132:
3109:
3099:
3095:
3092:of a nonzero
3091:
3087:
3071:
3067:
3063:
3055:
3049:
3045:
3037:
3035:
3033:
3029:
3025:
3021:
3018:
3014:
3010:
3007:
3003:
2999:
2961:
2955:
2939:
2935:
2931:
2928:
2923:
2921:
2917:
2913:
2909:
2905:
2902:, and indeed
2901:
2897:
2893:
2889:
2885:
2881:
2877:
2873:
2869:
2865:
2860:
2858:
2837:
2833:
2824:
2819:
2809:
2803:
2797:
2794:
2787:
2783:
2763:
2759:
2748:
2740:
2736:
2729:
2726:
2719:
2715:
2711:
2707:
2702:
2700:
2696:
2692:
2688:
2683:
2681:
2677:
2676:residue class
2673:
2669:
2650:
2647:
2642:
2638:
2630:
2623:
2617:
2614:
2611:
2603:
2599:
2595:
2591:
2575:
2572:
2567:
2563:
2554:
2550:
2546:
2542:
2538:
2534:
2530:
2527:
2523:
2522:quotient ring
2519:
2514:
2512:
2494:
2480:
2476:
2475:number theory
2472:
2427:
2423:
2414:
2411:
2408:
2405:
2402:
2399:
2396:
2393:
2390:
2385:
2375:
2370:
2365:
2357:
2354:
2349:
2339:
2334:
2329:
2321:
2318:
2310:
2305:
2300:
2292:
2289:
2286:
2282:
2278:
2276:
2263:
2258:
2253:
2240:
2238:
2228:
2223:
2218:
2197:
2196:
2195:
2181:
2151:
2114:
2110:
2101:
2098:
2095:
2092:
2089:
2086:
2083:
2080:
2077:
2072:
2067:
2064:
2059:
2054:
2051:
2046:
2041:
2038:
2035:
2031:
2027:
2025:
2016:
2011:
2006:
2001:
1992:
1989:
1986:
1983:
1980:
1975:
1970:
1967:
1964:
1960:
1956:
1954:
1945:
1940:
1935:
1930:
1925:
1920:
1911:
1909:
1903:
1897:
1892:
1887:
1881:
1864:
1863:
1862:
1859:
1844:
1838:
1833:
1830:
1826:
1805:
1777:
1773:
1764:
1761:
1758:
1755:
1752:
1747:
1742:
1739:
1736:
1732:
1728:
1720:
1703:
1702:
1701:
1698:
1696:
1673:
1662:
1635:
1629:
1618:
1587:
1558:
1555:
1552:
1529:
1526:
1515:
1481:
1402:
1372:
1370:
1367:
1365:
1361:
1357:
1353:
1349:
1346:
1340:
1338:
1334:
1330:
1326:
1325:quotient ring
1322:
1318:
1315:The notation
1310:
1308:
1306:
1302:
1298:
1294:
1290:
1285:
1281:
1277:
1271:
1269:
1265:
1260:
1258:
1254:
1250:
1240:
1236:
1230:
1228:
1224:
1220:
1215:
1211:
1207:
1202:
1198:
1194:
1184:
1180:
1164:
1153:
1149:
1145:
1142:
1139:
1134:
1130:
1120:
1095:
1091:
1087:
1084:
1081:
1076:
1072:
1065:
1040:
1036:
1032:
1029:
1026:
1021:
1017:
1010:
1007:
999:
995:
991:
987:
983:
979:
975:
971:
967:
963:
959:
956:
949:
945:
941:
937:
933:
930:that contain
929:
925:
921:
917:
913:
909:
906:and a subset
893:
889:
885:
862:
856:
853:
850:
844:
838:
835:
832:
826:
820:
817:
814:
804:
803:
802:
788:
784:
780:
760:
756:
752:
732:
728:
724:
704:
700:
696:
676:
672:
668:
659:
657:
653:
649:
645:
636:
619:
616:
613:
602:
600:
595:
591:
587:
583:
567:
563:
559:
550:
536:
532:
528:
520:
516:
512:
508:
504:
500:
496:
491:
489:
485:
469:
465:
461:
453:
449:
445:
441:
437:
433:
429:
421:
419:
417:
412:
398:
391:
387:
383:
379:
374:
360:
340:
327:
313:
305:
301:
285:
265:
245:
242:
239:
232:
216:
209:
193:
186:
178:
176:
174:
170:
169:Galois theory
166:
162:
157:
155:
151:
147:
143:
139:
135:
131:
127:
123:
119:
115:
111:
108:
105:are those of
104:
88:
85:
82:
75:
59:
55:
51:
43:
39:
35:
27:
19:
5407:
5377:
5355:
5330:
5308:
5287:
5275:
5263:
5251:
5239:
5227:
5215:
5203:
5170:Field theory
5135:
5084:
5062:
5058:
5054:
5024:
5012:Galois group
5007:
5003:
4999:
4995:
4991:
4987:
4983:
4979:
4934:
4904:
4897:
4888:
4880:
4876:
4842:
4809:
4805:
4801:
4797:
4793:
4789:
4785:
4747:
4727:
4057:
4052:
4048:
4044:
4040:
4036:
4032:
4028:
4024:
4020:
4016:
4014:
4009:
4005:
4001:
3997:
3993:
3989:
3985:
3981:
3977:
3973:
3941:
3933:
3901:
3897:
3892:is called a
3889:
3885:
3881:
3877:
3876:, such that
3873:
3869:
3865:
3861:
3853:
3849:
3845:
3837:
3833:
3803:
3673:
3669:
3665:
3657:
3656:Every field
3655:
3648:
3504:
3500:
3469:
3467:
3462:
3458:
3453:
3449:
3442:
3436:
3432:
3428:
3422:
3418:
3414:
3408:
3404:
3400:
3396:
3390:
3386:
3382:
3378:
3371:
3367:
3363:
3359:
3355:
3349:
3345:
3341:
3336:
3332:
3328:
3324:
3316:
3312:
3310:
3305:
3234:
3230:
3228:vector space
3223:
3218:
3214:
3210:
3205:
3201:
3197:
3195:
3190:
3182:
3174:
3166:
3162:
3158:
3097:
3085:
3053:
3051:
3031:
3027:
3023:
3019:
3012:
3008:
3001:
2937:
2929:
2924:
2919:
2915:
2907:
2903:
2899:
2891:
2887:
2883:
2879:
2871:
2867:
2863:
2861:
2856:
2781:
2718:finite field
2713:
2710:prime number
2705:
2703:
2698:
2694:
2690:
2684:
2679:
2671:
2667:
2593:
2552:
2548:
2544:
2540:
2536:
2528:
2517:
2515:
2510:
2446:
2133:
1860:
1792:
1699:
1401:real numbers
1376:
1368:
1351:
1341:
1336:
1332:
1320:
1316:
1314:
1304:
1300:
1292:
1288:
1283:
1279:
1275:
1272:
1261:
1256:
1252:
1244:
1238:
1234:
1231:
1225:is called a
1222:
1217:is called a
1213:
1209:
1205:
1200:
1196:
1192:
1182:
1178:
997:
993:
989:
981:
977:
973:
969:
965:
961:
957:
950:
947:
946:) (read as "
943:
939:
935:
931:
927:
923:
919:
915:
911:
907:
877:
660:
655:
651:
647:
639:
637:
598:
590:vector space
585:
581:
551:
519:subextension
518:
514:
510:
506:
502:
498:
494:
492:
487:
483:
451:
447:
443:
439:
435:
431:
427:
425:
413:
382:real numbers
375:
328:
184:
182:
158:
154:real numbers
137:
133:
129:
125:
121:
117:
113:
106:
102:
41:
31:
26:
5352:Lang, Serge
5333:, Waltham:
5232:McCoy (1968
5018:are called
4004:where both
3832:, a subset
3196:An element
3187:irreducible
3088:if it is a
3052:An element
3000:defined on
2936:defined on
2896:polynomials
2786:prime field
2590:irreducible
1700:The field
1113:instead of
980:, and that
34:mathematics
5380:, Boston:
5301:References
5120:local ring
4871:is called
4812:such that
4776:is called
4043:such that
3972:such that
3840:is called
3574:, because
3094:polynomial
2859:elements.
1861:The field
1247:adjunction
482:(read as "
442:or simply
386:isomorphic
110:restricted
5414:EMS Press
5071:subgroups
5067:bijection
4885:separable
4873:separable
4780:if every
4705:and thus
4300:in which
4285:⟩
4272:−
4259:⟨
4195:−
3284:−
3270:…
3142:−
2910:) is the
2874:) of all
2798:
2415:∈
2391:∣
2102:∈
2078:∣
1993:∈
1981:∣
1765:∈
1753:∣
1360:morphisms
1345:injective
1339:is used.
1143:…
1085:…
1030:…
970:generated
845:⋅
594:dimension
444:extension
331:1 – 1 = 0
298:, and is
243:⊆
86:⊆
44:(denoted
5425:Category
5390:68015225
5354:(2004),
5164:See also
5095:subrings
5002:for all
4636:one has
4340:are the
2925:Given a
1542:because
1373:Examples
185:subfield
179:Subfield
167:through
142:addition
134:subfield
5416:, 2001
5356:Algebra
5016:abelian
2708:is any
2602:maximal
2524:of the
1362:in the
1311:Caveats
1000:. When
996:) over
505:, then
434:, then
304:inverse
38:algebra
5388:
5366:
5341:
5319:
4778:normal
4108:where
3461:and 1/
3304:where
3169:, the
3015:, the
2670:which
2604:, and
1424:, and
1356:ideals
953:adjoin
599:degree
592:. The
438:is an
300:closed
231:subset
148:, the
120:is an
74:fields
5196:Notes
4990:with
4879:over
4173:is a
3844:over
3662:up to
3189:over
2898:over
2855:with
2780:with
2598:ideal
2551:with
2520:as a
1693:(the
1352:Every
1303:over
1195:. If
1191:over
1185:) is
984:is a
976:over
584:is a
521:) of
486:over
229:is a
208:field
206:of a
132:is a
5386:LCCN
5364:ISBN
5339:ISBN
5317:ISBN
5091:ring
4998:) =
4891:. A
4669:and
4364:and
4320:and
4175:root
4153:and
4008:and
3598:and
3440:(if
3426:and
3353:and
3335:and
3090:root
3046:and
2712:and
2672:does
2533:root
2169:and
1282:) /
1221:and
1212:) /
934:and
922:and
773:and
689:and
650:and
513:(or
490:").
414:The
353:and
144:and
128:and
40:, a
5006:in
4945:Aut
4883:is
4784:in
4732:of
4344:of
4055:).
4015:If
3944:of
3896:of
3860:of
3836:of
3790:).
3468:An
3445:≠ 0
3435:) /
3431:(1/
3421:) /
3407:) /
3389:) /
3366:) /
3348:) /
3327:in
3323:of
3217:) /
3200:of
3181:of
3161:of
2704:If
2697:of
2682:).
2678:of
2592:in
2588:is
1327:or
1299:in
1262:In
1255:to
1251:of
988:of
972:by
910:of
635:.
517:or
493:If
446:of
426:If
329:As
136:of
124:of
112:to
32:In
5427::
5412:,
5406:,
5384:,
5358:,
5337:,
5315:,
5160:.
5122:.
5077:.
5022:.
4986:→
4982::
4211:0.
4047:=
4031:=
3976:=
3459:st
3457:,
3452:±
3419:st
3412:,
3403:±
3385:,
3362:)(
3193:.
3145:2.
3034:.
2795:GF
2513:.
1366:.
1319:/
1307:.
1259:.
549:.
411:.
326:.
183:A
175:.
5063:K
5059:L
5055:F
5041:K
5037:/
5033:L
5008:K
5004:x
5000:x
4996:x
4994:(
4992:α
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4963:)
4960:K
4956:/
4952:L
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4760:/
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4713:t
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4688:3
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4677:y
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4621:x
4617:/
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4607:t
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4584:y
4581:{
4561:)
4558:x
4555:(
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4505:}
4502:x
4499:{
4479:)
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4395:{
4375:.
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4280:3
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4255:/
4251:]
4248:Y
4245:[
4242:)
4239:X
4236:(
4232:Q
4208:=
4203:3
4199:x
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4186:y
4161:y
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4096:,
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4087:/
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4074:x
4071:(
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4000:/
3998:L
3994:K
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3980:(
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3884:(
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3608:3
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3525:2
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3372:s
3370:(
3368:K
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3358:(
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3344:(
3342:K
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3333:s
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3306:d
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3277:s
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3267:,
3262:2
3258:s
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3231:K
3226:-
3224:K
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3215:s
3213:(
3211:K
3206:K
3202:L
3198:s
3191:K
3183:x
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3110:2
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3068:/
3064:L
3054:x
3032:K
3028:V
3024:V
3022:(
3020:K
3013:K
3009:V
3002:M
2983:C
2962:.
2959:)
2956:M
2953:(
2949:C
2938:M
2930:M
2920:K
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2890:(
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2872:X
2870:(
2868:K
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2834:/
2829:Z
2825:=
2820:p
2815:F
2810:=
2807:)
2804:p
2801:(
2782:p
2764:n
2760:p
2754:F
2749:=
2746:)
2741:n
2737:p
2733:(
2730:F
2727:G
2714:n
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2699:K
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2651:1
2648:+
2643:2
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2631:/
2627:]
2624:X
2621:[
2618:K
2615:=
2612:L
2594:K
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2573:+
2568:2
2564:X
2553:x
2549:x
2545:K
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2539:(
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2428:.
2424:}
2419:Q
2412:d
2409:,
2406:c
2403:,
2400:b
2397:,
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2386:3
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2371:+
2366:2
2361:(
2358:d
2355:+
2350:2
2346:)
2340:3
2335:+
2330:2
2325:(
2322:c
2319:+
2316:)
2311:3
2306:+
2301:2
2296:(
2293:b
2290:+
2287:a
2283:{
2279:=
2269:)
2264:3
2259:+
2254:2
2249:(
2245:Q
2241:=
2234:)
2229:3
2224:,
2219:2
2214:(
2210:Q
2182:,
2178:Q
2157:)
2152:2
2147:(
2143:Q
2115:,
2111:}
2106:Q
2099:d
2096:,
2093:c
2090:,
2087:b
2084:,
2081:a
2073:6
2068:d
2065:+
2060:3
2055:c
2052:+
2047:2
2042:b
2039:+
2036:a
2032:{
2028:=
2017:}
2012:)
2007:2
2002:(
1997:Q
1990:b
1987:,
1984:a
1976:3
1971:b
1968:+
1965:a
1961:{
1957:=
1946:)
1941:3
1936:(
1931:)
1926:2
1921:(
1916:Q
1912:=
1904:)
1898:3
1893:,
1888:2
1882:(
1877:Q
1845:}
1839:2
1834:,
1831:1
1827:{
1806:,
1802:Q
1778:,
1774:}
1769:Q
1762:b
1759:,
1756:a
1748:2
1743:b
1740:+
1737:a
1733:{
1729:=
1726:)
1721:2
1716:(
1712:Q
1679:c
1674:=
1671:]
1667:Q
1663::
1659:R
1655:[
1636:.
1633:)
1630:i
1627:(
1623:R
1619:=
1615:C
1593:R
1588:/
1583:C
1562:}
1559:i
1556:,
1553:1
1550:{
1530:2
1527:=
1524:]
1520:R
1516::
1512:C
1508:[
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1482:/
1477:C
1455:Q
1433:R
1411:R
1386:C
1337:K
1335::
1333:L
1321:K
1317:L
1305:K
1301:s
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1291:(
1289:K
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1276:K
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1239:S
1237:(
1235:K
1223:s
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1210:s
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1162:)
1159:}
1154:n
1150:x
1146:,
1140:,
1135:1
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1127:{
1124:(
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1101:)
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1046:}
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1033:,
1027:,
1022:1
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1014:{
1011:=
1008:S
998:K
994:S
992:(
990:K
982:S
978:K
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886:L
863:.
860:]
857:K
854::
851:L
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842:]
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836::
833:M
830:[
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824:]
821:K
818::
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812:[
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729:/
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611:[
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