1691:
50:
2245:
1446:
1844:
2052:
1033:
1686:{\displaystyle K\otimes L=K\otimes _{N}L=\mathbb {Q} /(x^{2}-2)\otimes _{\mathbb {Q} }\mathbb {Q} /(y^{2}-2)\cong \mathbb {Q} ({\sqrt {2}})/(z^{2}-2)\cong \mathbb {Q} ({\sqrt {2}})\otimes _{\mathbb {Q} }\mathbb {Q} ({\sqrt {2}})}
2044:
1920:
1410:
2810:
1724:
1970:
2668:
2578:
2314:
409:
1357:
1293:
720:
589:
2240:{\displaystyle K\otimes _{\tilde {N}}L=\mathbb {Q} /(x^{2}-2)\otimes _{\mathbb {Q} ({\sqrt {2}})}\mathbb {Q} /(y^{2}-2)\cong \mathbb {Q} ({\sqrt {2}})={\tilde {N}}\cong K\cong L}
1205:
916:
2507:
1438:
1108:
904:
793:
673:
505:
226:
2901:
2866:
2832:
2712:
2690:
2627:
2377:
2355:
2275:
1716:
752:
434:
367:
325:
157:. The idea behind the compositum is to make the smallest field containing two other fields. In order to formally define the compositum, one must first specify a
3096:
3040:
2046:. When one performs the tensor product over this better candidate for the largest common subfield we actually get a (rather trivial) field
1987:
3120:
1852:
2473:
1362:
1839:{\displaystyle K\otimes L\cong \mathbb {Q} ({\sqrt {2}})/(z^{2}-2)\cong \mathbb {Q} ({\sqrt {2}})\oplus \mathbb {Q} ({\sqrt {2}})}
2916:
2768:
796:
31:
1114:(intersection of all prime ideals) β and after taking the quotient by that one can speak of the product of all embeddings of
1925:
3019:
2991:
2967:
2636:
3135:
2515:
141:
The tensor product of two fields expresses in a single structure the different way to embed the two fields in a common
3014:
2986:
2962:
755:
437:
154:
108:
2957:
2715:
833:
2284:
379:
120:
1298:
1234:
678:
60:
549:
2600:
1412:
are isomorphic but technically unequal fields with their (set theoretic) intersection being the prime field
441:
116:
1028:{\displaystyle \gamma (a\otimes b)=(\alpha (a)\otimes 1)\star (1\otimes \beta (b))=\alpha (a).\beta (b).}
1163:
2981:
2936:
2433:
1111:
112:
2479:
1415:
1080:
876:
765:
645:
477:
184:
2920:
104:
2884:
2849:
2815:
2695:
2673:
2610:
2360:
2338:
2258:
1699:
725:
417:
350:
308:
153:
First, one defines the notion of the compositum of fields. This construction occurs frequently in
3065:
1059:
639:
3009:
75:
3092:
3036:
2417:
1208:
870:
460:
248:. Either one starts in a situation where an ambient field is easy to identify (for example if
135:
3028:
1148:, the situation is particularly simple since the tensor product is of finite dimension as an
71:
1047:
1039:
849:(thus getting round the caveats about constructing a compositum field). Whenever one embeds
611:
595:
303:
3106:
3102:
3088:
2457:
2321:
1055:
158:
142:
2754:
as one sees by counting dimensions. The field factors are in 1β1 correspondence with the
2379:
as 9, and observing that the splitting field does contain two (indeed three) copies of
1207:
as a direct product of finitely many fields. Each such field is a representative of an
603:
287:
257:
3129:
3080:
3076:
2912:
2840:
1153:
815:
The structure of the ring can be analysed by considering all ways of embedding both
2425:
284:
131:
38:
3051:
2719:
1043:
762:
in each variable; and so defines a ring structure on the tensor product, making
619:
531:
124:
96:
269:
17:
2357:. One can prove this by calculating the dimension of the tensor product over
2278:
519:
119:. If no subfield is explicitly specified, the two fields must have the same
1718:-algebra. Furthermore this algebra is isomorphic to a direct sum of fields
2939:βtensor product of a field extension and a vector space over that field
3066:"A Brief Introduction to Classical and Adelic Algebraic Number Theory"
2603:, tensor products of fields are (implicitly, often) a basic tool. If
2383:, and is the compositum of two of them. That incidentally shows that
2039:{\displaystyle {\tilde {N}}=\mathbb {Q} ({\sqrt {2}})\cong K\cong L}
2923:
the radical is always {0}; therefore the Galois theory case is the
130:
The tensor product of two fields is sometimes a field, and often a
1915:{\displaystyle 1\mapsto (1,1),z\mapsto ({\sqrt {2}},-{\sqrt {2}})}
1050:
any prime ideal of the tensor product will give a homomorphism of
370:
530:. When the degrees are finite, injectivity is equivalent here to
436:. (This type of result can be verified, in general, by using the
522:. Naturally enough this isn't always the case, for example when
43:
2911:
This gives a general picture, and indeed a way of developing
1405:{\displaystyle \mathbb {Q} ({\sqrt {2}})\cong K\cong L\neq K}
228:
where the right-hand side denotes the extension generated by
3121:
MathOverflow thread on the definition of linear disjointness
591:, as with the aforementioned extensions of the rationals.
542:
are linearly disjoint finite-degree extension fields over
2805:{\displaystyle K\otimes _{\mathbb {Q} }\mathbb {Q} _{p},}
260:), or one proves a result that allows one to place both
67:
1965:{\displaystyle {\tilde {N}}=\mathbb {Q} ({\sqrt {2}})}
2887:
2852:
2818:
2771:
2698:
2676:
2639:
2613:
2518:
2482:
2363:
2341:
2287:
2261:
2055:
1990:
1928:
1855:
1727:
1702:
1449:
1418:
1365:
1301:
1237:
1166:
1083:
919:
879:
768:
728:
681:
648:
552:
480:
420:
382:
353:
311:
187:
2663:{\displaystyle K\otimes _{\mathbb {Q} }\mathbb {R} }
827:. The construction here assumes the common subfield
2895:
2860:
2826:
2804:
2706:
2684:
2662:
2621:
2572:
2501:
2371:
2349:
2308:
2269:
2239:
2038:
1964:
1914:
1838:
1710:
1685:
1432:
1404:
1351:
1287:
1199:
1102:
1027:
898:
787:
746:
714:
667:
583:
499:
428:
403:
361:
319:
220:
134:of fields; In some cases, it can contain non-zero
2573:{\displaystyle T^{1/p}\otimes 1-1\otimes T^{1/p}}
638:To get a general theory, one needs to consider a
3029:"9.2 Decomposition of Tensor Products of Fields"
2873:, in 1β1 correspondence with the completions of
2390:An example leading to a non-zero nilpotent: let
1231:To give an explicit example consider the fields
3087:. Graduate Texts in Mathematics. Vol. 28.
2595:Classical theory of real and complex embeddings
1211:of (essentially distinct) field embeddings for
1077:In this way one can analyse the structure of
8:
2846:. This is a product of finite extensions of
2670:is always a product of fields isomorphic to
1070:in some field as extensions of (a copy of)
2889:
2888:
2886:
2854:
2853:
2851:
2820:
2819:
2817:
2793:
2789:
2788:
2781:
2780:
2779:
2770:
2762:, described in the classical literature.
2700:
2699:
2697:
2678:
2677:
2675:
2656:
2655:
2649:
2648:
2647:
2638:
2615:
2614:
2612:
2560:
2556:
2527:
2523:
2517:
2490:
2481:
2365:
2364:
2362:
2343:
2342:
2340:
2309:{\displaystyle K\otimes _{\mathbb {Q} }K}
2297:
2296:
2295:
2286:
2263:
2262:
2260:
2214:
2213:
2200:
2193:
2192:
2174:
2162:
2149:
2148:
2136:
2129:
2128:
2127:
2108:
2096:
2083:
2082:
2064:
2063:
2054:
2014:
2007:
2006:
1992:
1991:
1989:
1974:largest common subfield up to isomorphism
1952:
1945:
1944:
1930:
1929:
1927:
1902:
1889:
1854:
1826:
1819:
1818:
1805:
1798:
1797:
1779:
1767:
1748:
1741:
1740:
1726:
1704:
1703:
1701:
1673:
1666:
1665:
1659:
1658:
1657:
1643:
1636:
1635:
1617:
1605:
1586:
1579:
1578:
1560:
1548:
1535:
1534:
1528:
1527:
1526:
1507:
1495:
1482:
1481:
1469:
1448:
1426:
1425:
1417:
1374:
1367:
1366:
1364:
1334:
1322:
1309:
1308:
1300:
1270:
1258:
1245:
1244:
1236:
1189:
1177:
1165:
1091:
1082:
918:
887:
878:
776:
767:
727:
680:
656:
647:
572:
551:
488:
479:
422:
421:
419:
404:{\displaystyle K\otimes _{\mathbb {Q} }L}
392:
391:
390:
381:
355:
354:
352:
313:
312:
310:
186:
1352:{\displaystyle L=\mathbb {Q} /(y^{2}-2)}
1288:{\displaystyle K=\mathbb {Q} /(x^{2}-2)}
715:{\displaystyle (a\otimes b)(c\otimes d)}
302:. For example, if one adjoins β2 to the
2949:
1110:: there may in principle be a non-zero
584:{\displaystyle K.L\cong K\otimes _{N}L}
2760:pairs of complex conjugate embeddings
7:
1696:is not a field, but a 4-dimensional
758:). This formula is multilinear over
594:A significant case in the theory of
272:copies) in some large enough field.
2722:fields occur: in general there are
626:are linearly disjoint for distinct
74:in tone and meet Knowledge (XXG)'s
2927:one, of products of fields alone.
2474:purely inseparable field extension
1200:{\displaystyle (K\otimes _{N}L)/R}
25:
614:, the subfields generated by the
123:and the common subfield is their
1062:) and so provides embeddings of
48:
467:) when in this way the natural
275:In many cases one can identify
32:Tensor product (disambiguation)
2958:"Linearly-disjoint extensions"
2907:Consequences for Galois theory
2502:{\displaystyle L\otimes _{K}L}
2219:
2207:
2197:
2186:
2167:
2159:
2153:
2143:
2133:
2120:
2101:
2093:
2087:
2069:
2021:
2011:
1997:
1959:
1949:
1935:
1909:
1886:
1883:
1874:
1862:
1859:
1833:
1823:
1812:
1802:
1791:
1772:
1764:
1758:
1755:
1745:
1680:
1670:
1650:
1640:
1629:
1610:
1602:
1596:
1593:
1583:
1572:
1553:
1545:
1539:
1519:
1500:
1492:
1486:
1433:{\displaystyle N=\mathbb {Q} }
1381:
1371:
1346:
1327:
1319:
1313:
1282:
1263:
1255:
1249:
1186:
1167:
1103:{\displaystyle K\otimes _{N}L}
1019:
1013:
1004:
998:
989:
986:
980:
968:
962:
953:
947:
941:
935:
923:
899:{\displaystyle K\otimes _{N}L}
811:Analysis of the ring structure
788:{\displaystyle K\otimes _{N}L}
709:
697:
694:
682:
668:{\displaystyle K\otimes _{N}L}
500:{\displaystyle K\otimes _{N}L}
221:{\displaystyle K.L=k(K\cup L)}
215:
203:
1:
3050:Milne, J.S. (18 March 2017).
2587:th power one gets 0 by using
675:. One can define the product
618: th roots of unity for
27:Ring produced from two fields
3035:. Springer. pp. 85β87.
2919:). It can be shown that for
2917:Grothendieck's Galois theory
2896:{\displaystyle \mathbb {Q} }
2861:{\displaystyle \mathbb {Q} }
2827:{\displaystyle \mathbb {Q} }
2707:{\displaystyle \mathbb {C} }
2685:{\displaystyle \mathbb {R} }
2622:{\displaystyle \mathbb {Q} }
2583:is nilpotent: by taking its
2372:{\displaystyle \mathbb {Q} }
2350:{\displaystyle \mathbb {Q} }
2270:{\displaystyle \mathbb {Q} }
1711:{\displaystyle \mathbb {Q} }
1156:). One can then say that if
861:, say using embeddings Ξ± of
845:are subfields of some field
747:{\displaystyle ac\otimes bd}
429:{\displaystyle \mathbb {Q} }
362:{\displaystyle \mathbb {C} }
335:, it is true that the field
320:{\displaystyle \mathbb {Q} }
294:that is the intersection of
3015:Encyclopedia of Mathematics
2987:Encyclopedia of Mathematics
2963:Encyclopedia of Mathematics
1046:of the tensor product; and
823:in some field extension of
347:inside the complex numbers
3152:
3027:Kempf, George R. (2012) .
2915:(along lines exploited in
2765:This idea applies also to
2716:totally real number fields
2316:is the sum of (a copy of)
756:Tensor product of algebras
634:The tensor product as ring
256:are both subfields of the
177:. The compositum, denoted
36:
29:
2718:are those for which only
2436:: the point here is that
1972:should be considered the
1144:are finite extensions of
2251:For another example, if
1160:is the radical, one has
805:tensor product of fields
37:Not to be confused with
3064:Stein, William (2004).
3053:Algebraic Number Theory
2601:algebraic number theory
2448:is the field extension
1849:via the map induced by
1440:. Their tensor product
442:algebraic number theory
414:as a vector space over
290:, taken over the field
2897:
2877:for extensions of the
2862:
2828:
2806:
2708:
2686:
2664:
2623:
2574:
2503:
2373:
2351:
2310:
2271:
2241:
2040:
1966:
1916:
1840:
1712:
1687:
1434:
1406:
1353:
1289:
1201:
1152:-algebra (and thus an
1104:
1029:
900:
831:; but does not assume
789:
748:
716:
669:
585:
501:
430:
405:
363:
321:
240:field containing both
222:
3085:Commutative algebra I
2898:
2863:
2829:
2807:
2736:complex fields, with
2709:
2687:
2665:
2624:
2575:
2504:
2420:in the indeterminate
2374:
2352:
2311:
2272:
2242:
2041:
1984:via the isomorphisms
1967:
1917:
1841:
1713:
1688:
1435:
1407:
1354:
1290:
1202:
1105:
1030:
901:
790:
749:
717:
670:
586:
502:
431:
406:
364:
322:
223:
173:be two extensions of
3033:Algebraic Structures
2937:Extension of scalars
2921:separable extensions
2885:
2850:
2816:
2769:
2696:
2674:
2637:
2611:
2516:
2480:
2434:Separable polynomial
2387:= {0} in this case.
2361:
2339:
2285:
2259:
2053:
1988:
1926:
1853:
1725:
1700:
1447:
1416:
1363:
1299:
1235:
1164:
1081:
917:
877:
766:
726:
679:
646:
550:
478:
418:
380:
351:
309:
185:
149:Compositum of fields
68:improve this article
30:For other uses, see
3136:Field (mathematics)
3059:. p. 17. 3.07.
2607:is an extension of
2472:is an example of a
181:, is defined to be
2982:"Cyclotomic field"
2893:
2858:
2824:
2802:
2704:
2682:
2660:
2619:
2570:
2499:
2418:rational functions
2369:
2347:
2306:
2267:
2255:is generated over
2237:
2036:
1962:
1912:
1836:
1708:
1683:
1430:
1402:
1349:
1285:
1219:in some extension
1197:
1100:
1060:field of fractions
1025:
896:
869:, there results a
785:
744:
712:
665:
581:
497:
426:
401:
359:
317:
218:
136:nilpotent elements
3098:978-0-387-90089-6
3071:. pp. 140β2.
3042:978-3-322-80278-1
2629:of finite degree
2335:of degree 6 over
2222:
2205:
2141:
2072:
2019:
2000:
1957:
1938:
1907:
1894:
1831:
1810:
1753:
1678:
1648:
1591:
1379:
1209:equivalence class
871:ring homomorphism
596:cyclotomic fields
463:(over a subfield
461:linearly disjoint
93:
92:
76:quality standards
16:(Redirected from
3143:
3110:
3072:
3070:
3060:
3058:
3046:
3023:
2996:
2995:
2978:
2972:
2971:
2954:
2902:
2900:
2899:
2894:
2892:
2881:-adic metric on
2867:
2865:
2864:
2859:
2857:
2839:is the field of
2833:
2831:
2830:
2825:
2823:
2811:
2809:
2808:
2803:
2798:
2797:
2792:
2786:
2785:
2784:
2713:
2711:
2710:
2705:
2703:
2691:
2689:
2688:
2683:
2681:
2669:
2667:
2666:
2661:
2659:
2654:
2653:
2652:
2628:
2626:
2625:
2620:
2618:
2579:
2577:
2576:
2571:
2569:
2568:
2564:
2536:
2535:
2531:
2508:
2506:
2505:
2500:
2495:
2494:
2378:
2376:
2375:
2370:
2368:
2356:
2354:
2353:
2348:
2346:
2315:
2313:
2312:
2307:
2302:
2301:
2300:
2276:
2274:
2273:
2268:
2266:
2246:
2244:
2243:
2238:
2224:
2223:
2215:
2206:
2201:
2196:
2179:
2178:
2166:
2152:
2147:
2146:
2142:
2137:
2132:
2113:
2112:
2100:
2086:
2075:
2074:
2073:
2065:
2045:
2043:
2042:
2037:
2020:
2015:
2010:
2002:
2001:
1993:
1971:
1969:
1968:
1963:
1958:
1953:
1948:
1940:
1939:
1931:
1921:
1919:
1918:
1913:
1908:
1903:
1895:
1890:
1845:
1843:
1842:
1837:
1832:
1827:
1822:
1811:
1806:
1801:
1784:
1783:
1771:
1754:
1749:
1744:
1717:
1715:
1714:
1709:
1707:
1692:
1690:
1689:
1684:
1679:
1674:
1669:
1664:
1663:
1662:
1649:
1644:
1639:
1622:
1621:
1609:
1592:
1587:
1582:
1565:
1564:
1552:
1538:
1533:
1532:
1531:
1512:
1511:
1499:
1485:
1474:
1473:
1439:
1437:
1436:
1431:
1429:
1411:
1409:
1408:
1403:
1380:
1375:
1370:
1358:
1356:
1355:
1350:
1339:
1338:
1326:
1312:
1294:
1292:
1291:
1286:
1275:
1274:
1262:
1248:
1206:
1204:
1203:
1198:
1193:
1182:
1181:
1109:
1107:
1106:
1101:
1096:
1095:
1054:-algebras to an
1034:
1032:
1031:
1026:
905:
903:
902:
897:
892:
891:
857:in such a field
794:
792:
791:
786:
781:
780:
753:
751:
750:
745:
721:
719:
718:
713:
674:
672:
671:
666:
661:
660:
612:composite number
598:is that for the
590:
588:
587:
582:
577:
576:
506:
504:
503:
498:
493:
492:
435:
433:
432:
427:
425:
410:
408:
407:
402:
397:
396:
395:
368:
366:
365:
360:
358:
331:, and β3 to get
326:
324:
323:
318:
316:
227:
225:
224:
219:
88:
85:
79:
52:
51:
44:
21:
3151:
3150:
3146:
3145:
3144:
3142:
3141:
3140:
3126:
3125:
3117:
3099:
3089:Springer-Verlag
3075:
3068:
3063:
3056:
3049:
3043:
3026:
3008:
3005:
3000:
2999:
2980:
2979:
2975:
2956:
2955:
2951:
2946:
2933:
2909:
2883:
2882:
2872:
2848:
2847:
2838:
2814:
2813:
2787:
2775:
2767:
2766:
2756:real embeddings
2749:
2743: + 2
2742:
2735:
2728:
2694:
2693:
2672:
2671:
2643:
2635:
2634:
2609:
2608:
2597:
2552:
2519:
2514:
2513:
2486:
2478:
2477:
2458:splitting field
2444:separable). If
2359:
2358:
2337:
2336:
2322:splitting field
2291:
2283:
2282:
2257:
2256:
2170:
2123:
2104:
2059:
2051:
2050:
1986:
1985:
1924:
1923:
1851:
1850:
1775:
1723:
1722:
1698:
1697:
1653:
1613:
1556:
1522:
1503:
1465:
1445:
1444:
1414:
1413:
1361:
1360:
1330:
1297:
1296:
1266:
1233:
1232:
1229:
1173:
1162:
1161:
1087:
1079:
1078:
1056:integral domain
1042:of Ξ³ will be a
915:
914:
883:
875:
874:
813:
772:
764:
763:
724:
723:
677:
676:
652:
644:
643:
636:
568:
548:
547:
484:
476:
475:
471:-linear map of
416:
415:
386:
378:
377:
349:
348:
307:
306:
258:complex numbers
236:. This assumes
183:
182:
165:be a field and
159:tower of fields
151:
143:extension field
89:
83:
80:
65:
53:
49:
42:
35:
28:
23:
22:
15:
12:
11:
5:
3149:
3147:
3139:
3138:
3128:
3127:
3124:
3123:
3116:
3115:External links
3113:
3112:
3111:
3097:
3081:Samuel, Pierre
3077:Zariski, Oscar
3073:
3061:
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3041:
3024:
3004:
3001:
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2891:
2868:
2856:
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2796:
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2747:
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2642:
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2596:
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2559:
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2548:
2545:
2542:
2539:
2534:
2530:
2526:
2522:
2498:
2493:
2489:
2485:
2432:elements (see
2410:
2409:
2367:
2345:
2333:
2332:
2305:
2299:
2294:
2290:
2265:
2249:
2248:
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2218:
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2209:
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2199:
2195:
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2188:
2185:
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2173:
2169:
2165:
2161:
2158:
2155:
2151:
2145:
2140:
2135:
2131:
2126:
2122:
2119:
2116:
2111:
2107:
2103:
2099:
2095:
2092:
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2085:
2081:
2078:
2071:
2068:
2062:
2058:
2035:
2032:
2029:
2026:
2023:
2018:
2013:
2009:
2005:
1999:
1996:
1961:
1956:
1951:
1947:
1943:
1937:
1934:
1911:
1906:
1901:
1898:
1893:
1888:
1885:
1882:
1879:
1876:
1873:
1870:
1867:
1864:
1861:
1858:
1847:
1846:
1835:
1830:
1825:
1821:
1817:
1814:
1809:
1804:
1800:
1796:
1793:
1790:
1787:
1782:
1778:
1774:
1770:
1766:
1763:
1760:
1757:
1752:
1747:
1743:
1739:
1736:
1733:
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1706:
1694:
1693:
1682:
1677:
1672:
1668:
1661:
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1652:
1647:
1642:
1638:
1634:
1631:
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1625:
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1608:
1604:
1601:
1598:
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1547:
1544:
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1537:
1530:
1525:
1521:
1518:
1515:
1510:
1506:
1502:
1498:
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1480:
1477:
1472:
1468:
1464:
1461:
1458:
1455:
1452:
1428:
1424:
1421:
1401:
1398:
1395:
1392:
1389:
1386:
1383:
1378:
1373:
1369:
1348:
1345:
1342:
1337:
1333:
1329:
1325:
1321:
1318:
1315:
1311:
1307:
1304:
1284:
1281:
1278:
1273:
1269:
1265:
1261:
1257:
1254:
1251:
1247:
1243:
1240:
1228:
1225:
1196:
1192:
1188:
1185:
1180:
1176:
1172:
1169:
1099:
1094:
1090:
1086:
1036:
1035:
1024:
1021:
1018:
1015:
1012:
1009:
1006:
1003:
1000:
997:
994:
991:
988:
985:
982:
979:
976:
973:
970:
967:
964:
961:
958:
955:
952:
949:
946:
943:
940:
937:
934:
931:
928:
925:
922:
895:
890:
886:
882:
812:
809:
784:
779:
775:
771:
743:
740:
737:
734:
731:
711:
708:
705:
702:
699:
696:
693:
690:
687:
684:
664:
659:
655:
651:
635:
632:
604:roots of unity
580:
575:
571:
567:
564:
561:
558:
555:
534:. Hence, when
508:
507:
496:
491:
487:
483:
424:
412:
411:
400:
394:
389:
385:
357:
315:
304:rational field
288:tensor product
217:
214:
211:
208:
205:
202:
199:
196:
193:
190:
150:
147:
132:direct product
125:prime subfield
121:characteristic
115:over a common
109:tensor product
101:tensor product
91:
90:
56:
54:
47:
26:
24:
18:Real embedding
14:
13:
10:
9:
6:
4:
3:
2:
3148:
3137:
3134:
3133:
3131:
3122:
3119:
3118:
3114:
3108:
3104:
3100:
3094:
3090:
3086:
3082:
3078:
3074:
3067:
3062:
3055:
3054:
3048:
3044:
3038:
3034:
3030:
3025:
3021:
3017:
3016:
3011:
3007:
3006:
3002:
2993:
2989:
2988:
2983:
2977:
2974:
2969:
2965:
2964:
2959:
2953:
2950:
2943:
2938:
2935:
2934:
2930:
2928:
2926:
2922:
2918:
2914:
2913:Galois theory
2906:
2904:
2880:
2876:
2871:
2845:
2844:-adic numbers
2843:
2837:
2799:
2794:
2776:
2772:
2763:
2761:
2757:
2753:
2746:
2739:
2732:
2725:
2721:
2717:
2644:
2640:
2632:
2606:
2602:
2594:
2592:
2590:
2586:
2565:
2561:
2557:
2553:
2549:
2546:
2543:
2540:
2537:
2532:
2528:
2524:
2520:
2512:
2511:
2510:
2496:
2491:
2487:
2483:
2475:
2471:
2467:
2463:
2459:
2455:
2451:
2447:
2443:
2439:
2435:
2431:
2427:
2423:
2419:
2416:the field of
2415:
2408:
2404:
2400:
2396:
2393:
2392:
2391:
2388:
2386:
2382:
2330:
2327:
2326:
2325:
2323:
2319:
2303:
2292:
2288:
2280:
2254:
2234:
2231:
2228:
2225:
2216:
2210:
2202:
2189:
2183:
2180:
2175:
2171:
2163:
2156:
2138:
2124:
2117:
2114:
2109:
2105:
2097:
2090:
2079:
2076:
2066:
2060:
2056:
2049:
2048:
2047:
2033:
2030:
2027:
2024:
2016:
2003:
1994:
1983:
1979:
1975:
1954:
1941:
1932:
1904:
1899:
1896:
1891:
1880:
1877:
1871:
1868:
1865:
1856:
1828:
1815:
1807:
1794:
1788:
1785:
1780:
1776:
1768:
1761:
1750:
1737:
1734:
1731:
1728:
1721:
1720:
1719:
1675:
1654:
1645:
1632:
1626:
1623:
1618:
1614:
1606:
1599:
1588:
1575:
1569:
1566:
1561:
1557:
1549:
1542:
1523:
1516:
1513:
1508:
1504:
1496:
1489:
1478:
1475:
1470:
1466:
1462:
1459:
1456:
1453:
1450:
1443:
1442:
1441:
1422:
1419:
1399:
1396:
1393:
1390:
1387:
1384:
1376:
1343:
1340:
1335:
1331:
1323:
1316:
1305:
1302:
1279:
1276:
1271:
1267:
1259:
1252:
1241:
1238:
1226:
1224:
1222:
1218:
1214:
1210:
1194:
1190:
1183:
1178:
1174:
1170:
1159:
1155:
1154:Artinian ring
1151:
1147:
1143:
1139:
1134:
1132:
1129:
1125:
1121:
1117:
1113:
1097:
1092:
1088:
1084:
1075:
1073:
1069:
1065:
1061:
1057:
1053:
1049:
1045:
1041:
1022:
1016:
1010:
1007:
1001:
995:
992:
983:
977:
974:
971:
965:
959:
956:
950:
944:
938:
932:
929:
926:
920:
913:
912:
911:
909:
893:
888:
884:
880:
872:
868:
864:
860:
856:
852:
848:
844:
840:
836:
835:
830:
826:
822:
818:
810:
808:
806:
803:, called the
802:
800:
782:
777:
773:
769:
761:
757:
741:
738:
735:
732:
729:
706:
703:
700:
691:
688:
685:
662:
657:
653:
649:
642:structure on
641:
633:
631:
629:
625:
621:
617:
613:
609:
605:
601:
597:
592:
578:
573:
569:
565:
562:
559:
556:
553:
545:
541:
537:
533:
529:
525:
521:
517:
513:
494:
489:
485:
481:
474:
473:
472:
470:
466:
462:
458:
454:
450:
445:
443:
439:
398:
387:
383:
376:
375:
374:
373:isomorphism)
372:
346:
342:
338:
334:
330:
305:
301:
297:
293:
289:
286:
282:
278:
273:
271:
267:
263:
259:
255:
251:
247:
243:
239:
235:
231:
212:
209:
206:
200:
197:
194:
191:
188:
180:
176:
172:
168:
164:
160:
156:
148:
146:
144:
139:
137:
133:
128:
126:
122:
118:
114:
110:
106:
102:
98:
87:
77:
73:
69:
63:
62:
59:reads like a
57:This article
55:
46:
45:
40:
33:
19:
3084:
3052:
3032:
3013:
3010:"Compositum"
2985:
2976:
2961:
2952:
2924:
2910:
2878:
2874:
2869:
2841:
2835:
2764:
2759:
2755:
2751:
2744:
2737:
2730:
2723:
2630:
2604:
2598:
2591:-linearity.
2588:
2584:
2582:
2509:the element
2469:
2465:
2461:
2453:
2449:
2445:
2441:
2437:
2429:
2426:finite field
2421:
2413:
2411:
2406:
2402:
2398:
2394:
2389:
2384:
2380:
2334:
2328:
2317:
2252:
2250:
1981:
1977:
1973:
1848:
1695:
1230:
1220:
1216:
1212:
1157:
1149:
1145:
1141:
1137:
1135:
1130:
1127:
1123:
1119:
1115:
1076:
1071:
1067:
1063:
1051:
1037:
910:defined by:
907:
866:
862:
858:
854:
850:
846:
842:
838:
832:
828:
824:
820:
816:
814:
804:
798:
797:commutative
759:
637:
627:
623:
620:prime powers
615:
607:
599:
593:
543:
539:
535:
527:
523:
515:
511:
509:
468:
464:
456:
452:
448:
446:
438:ramification
413:
344:
340:
339:obtained as
336:
332:
328:
299:
295:
291:
285:vector space
280:
276:
274:
265:
261:
253:
249:
245:
241:
237:
233:
229:
178:
174:
170:
166:
162:
155:field theory
152:
140:
129:
100:
94:
81:
58:
39:Tensor field
2281:of 2, then
1122:in various
1044:prime ideal
532:bijectivity
97:mathematics
84:August 2021
70:to make it
3003:References
2925:semisimple
1922:. Morally
1359:. Clearly
1112:nilradical
1058:(inside a
1048:conversely
447:Subfields
440:theory of
270:isomorphic
3083:(1975) .
3020:EMS Press
2992:EMS Press
2968:EMS Press
2777:⊗
2729:real and
2645:⊗
2550:⊗
2544:−
2538:⊗
2488:⊗
2424:over the
2293:⊗
2279:cube root
2232:≅
2226:≅
2220:~
2190:≅
2181:−
2125:⊗
2115:−
2070:~
2061:⊗
2031:≅
2025:≅
1998:~
1936:~
1900:−
1884:↦
1860:↦
1816:⊕
1795:≅
1786:−
1738:≅
1732:⊗
1655:⊗
1633:≅
1624:−
1576:≅
1567:−
1524:⊗
1514:−
1467:⊗
1454:⊗
1397:≠
1391:≅
1385:≅
1341:−
1277:−
1175:⊗
1089:⊗
1011:β
996:α
978:β
975:⊗
966:⋆
957:⊗
945:α
930:⊗
921:γ
885:⊗
865:and Ξ² of
774:⊗
736:⊗
704:⊗
689:⊗
654:⊗
622:dividing
570:⊗
563:≅
520:injective
486:⊗
388:⊗
210:∪
107:is their
3130:Category
2931:See also
2320:, and a
1227:Examples
1136:In case
834:a priori
801:-algebra
117:subfield
113:algebras
61:textbook
3107:0090581
3022:, 2001
2994:, 2001
2970:, 2001
2464:) then
2456:) (the
2277:by the
873:Ξ³ from
795:into a
327:to get
103:of two
72:neutral
66:Please
3105:
3095:
3039:
2812:where
2758:, and
2714:. The
1040:kernel
722:to be
606:, for
161:. Let
105:fields
99:, the
3069:(PDF)
3057:(PDF)
2944:Notes
2476:. In
2428:with
2412:with
906:into
837:that
754:(see
371:up to
283:as a
3093:ISBN
3037:ISBN
2720:real
2401:) =
2331:β 2,
1980:and
1295:and
1215:and
1140:and
1128:over
1118:and
1066:and
1038:The
853:and
841:and
819:and
640:ring
538:and
459:are
451:and
369:is (
298:and
268:(as
264:and
252:and
244:and
238:some
232:and
169:and
2692:or
2599:In
2460:of
2442:not
2440:is
2324:of
1976:of
602:th
518:is
510:to
455:of
444:.)
179:K.L
111:as
95:In
3132::
3103:MR
3101:.
3091:.
3079:;
3031:.
3018:,
3012:,
2990:,
2984:,
2966:,
2960:,
2903:.
2750:=
2633:,
2405:β
1223:.
1133:.
1126:,
1074:.
807:.
630:.
610:a
546:,
526:=
145:.
138:.
127:.
3109:.
3045:.
2890:Q
2879:p
2875:K
2870:p
2855:Q
2842:p
2836:p
2821:Q
2800:,
2795:p
2790:Q
2782:Q
2773:K
2752:n
2748:2
2745:r
2741:1
2738:r
2734:2
2731:r
2727:1
2724:r
2701:C
2679:R
2657:R
2650:Q
2641:K
2631:n
2616:Q
2605:K
2589:K
2585:p
2566:p
2562:/
2558:1
2554:T
2547:1
2541:1
2533:p
2529:/
2525:1
2521:T
2497:L
2492:K
2484:L
2470:K
2468:/
2466:L
2462:P
2454:T
2452:(
2450:K
2446:L
2438:P
2430:p
2422:T
2414:K
2407:T
2403:X
2399:X
2397:(
2395:P
2385:R
2381:K
2366:Q
2344:Q
2329:X
2318:K
2304:K
2298:Q
2289:K
2264:Q
2253:K
2247:.
2235:L
2229:K
2217:N
2211:=
2208:)
2203:2
2198:(
2194:Q
2187:)
2184:2
2176:2
2172:y
2168:(
2164:/
2160:]
2157:y
2154:[
2150:Q
2144:)
2139:2
2134:(
2130:Q
2121:)
2118:2
2110:2
2106:x
2102:(
2098:/
2094:]
2091:x
2088:[
2084:Q
2080:=
2077:L
2067:N
2057:K
2034:L
2028:K
2022:)
2017:2
2012:(
2008:Q
2004:=
1995:N
1982:L
1978:K
1960:)
1955:2
1950:(
1946:Q
1942:=
1933:N
1910:)
1905:2
1897:,
1892:2
1887:(
1881:z
1878:,
1875:)
1872:1
1869:,
1866:1
1863:(
1857:1
1834:)
1829:2
1824:(
1820:Q
1813:)
1808:2
1803:(
1799:Q
1792:)
1789:2
1781:2
1777:z
1773:(
1769:/
1765:]
1762:z
1759:[
1756:)
1751:2
1746:(
1742:Q
1735:L
1729:K
1705:Q
1681:)
1676:2
1671:(
1667:Q
1660:Q
1651:)
1646:2
1641:(
1637:Q
1630:)
1627:2
1619:2
1615:z
1611:(
1607:/
1603:]
1600:z
1597:[
1594:)
1589:2
1584:(
1580:Q
1573:)
1570:2
1562:2
1558:y
1554:(
1550:/
1546:]
1543:y
1540:[
1536:Q
1529:Q
1520:)
1517:2
1509:2
1505:x
1501:(
1497:/
1493:]
1490:x
1487:[
1483:Q
1479:=
1476:L
1471:N
1463:K
1460:=
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1427:Q
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1420:N
1400:K
1394:L
1388:K
1382:)
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1372:(
1368:Q
1347:)
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1332:y
1328:(
1324:/
1320:]
1317:y
1314:[
1310:Q
1306:=
1303:L
1283:)
1280:2
1272:2
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1264:(
1260:/
1256:]
1253:x
1250:[
1246:Q
1242:=
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1195:R
1191:/
1187:)
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1179:N
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1168:(
1158:R
1150:N
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1052:N
1023:.
1020:)
1017:b
1014:(
1008:.
1005:)
1002:a
999:(
993:=
990:)
987:)
984:b
981:(
972:1
969:(
963:)
960:1
954:)
951:a
948:(
942:(
939:=
936:)
933:b
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908:M
894:L
889:N
881:K
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863:K
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778:N
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760:N
742:d
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733:c
730:a
710:)
707:d
701:c
698:(
695:)
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683:(
663:L
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628:p
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579:L
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560:L
557:.
554:K
544:N
540:L
536:K
528:L
524:K
516:L
514:.
512:K
495:L
490:N
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469:N
465:N
457:M
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449:K
423:Q
399:L
393:Q
384:K
356:C
345:L
343:.
341:K
337:M
333:L
329:K
314:Q
300:L
296:K
292:N
281:L
279:.
277:K
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254:L
250:K
246:L
242:K
234:L
230:K
216:)
213:L
207:K
204:(
201:k
198:=
195:L
192:.
189:K
175:k
171:K
167:L
163:k
86:)
82:(
78:.
64:.
41:.
34:.
20:)
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