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Galois representation

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Ordinary representations. These are related to the representations of elliptic curves with ordinary (non-supersingular) reduction. More precisely, they are 2-dimensional representations that are reducible with a 1-dimensional subrepresentation, such that the inertia group acts in a certain way on the
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There are numerous conditions on representations given by some property of the representation restricted to a decomposition group of some prime. The terminology for these conditions is somewhat chaotic, with different authors inventing different names for the same condition and using the same name
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Finite flat representations. (This name is a little misleading, as they are really profinite rather than finite.) These can be constructed as a projective limit of representations of the Galois group on a finite flat
1147: 880: 836:. Other examples come from the Galois representations of modular forms and automorphic forms, and the Galois representations on ℓ-adic cohomology groups of algebraic varieties. 1510: 991:
submodule and the quotient. The exact condition depends on the author; for example it might act trivially on the quotient and by the character Δ on the submodule.
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There are also Galois representations that arise from auxiliary objects and can be used to study Galois groups. An important family of examples are the
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These are representations over a finite field of characteristic ℓ. They often arise as the reduction mod ℓ of an ℓ-adic representation.
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representations. This means that the representations restricted to an open subgroup of finite index has some specified property.
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module. It leaves the question of the gap between free and projective, for which a large theory has now been built up.
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Semistable representations. These are two dimensional representations related to the representations coming from
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is a local field, the multiplicative group of its separable closure is a module for the absolute Galois group of
1571: 690: 1814:, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, vol. 1, Berlin-Heidelberg-New York-Tokyo: 1782: 984: 908: 882:. ℓ-adic representations with finite image are often called Artin representations. Via an isomorphism of 829: 543: 455: 839:
Unlike Artin representations, ℓ-adic representations can have infinite image. For example, the image of
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Abelian representations. This means that the image of the Galois group in the representations is
694: 274: 144: 1692:, Proc. Sympos. Pure Math., vol. 55, Providence, R.I.: Amer. Math. Soc., pp. 365–392, 973:. These are irreducible in the sense that the only subrepresentation is the whole space or zero. 1854: 1491: 554:
Many objects that arise in number theory are naturally Galois representations. For example, if
315:-module, and one can ask what its structure is. This is an arithmetic question, in that by the 1756:, Proc. Sympos. Pure Math., vol. 33, Providence, R.I.: Amer. Math. Soc., pp. 3–26, 686: 124: 71: 1732: 682: 539: 400:), is there a normal integral basis? The answer is yes, as one sees by identifying it with 316: 161: 1833: 1807: 1800: 1740: 1688:
Kudla, Stephen S. (1994), "The local Langlands correspondence: the non-archimedean case",
1334: â‰  â„“, then it is simpler to study the so-called Weil–Deligne representations of 8: 1024:
Wildly ramified representations. These are non-trivial on the (first) ramification group.
720: 671: 615: 47: 1301: 1012: 747: 709: 512:. Then Noether's theorem states that tame ramification is necessary and sufficient for 67: 1706: 1819: 1786: 1757: 1718: 1693: 1421: 1077: 934: 611: 520: 459: 345: 299: 169: 116: 87: 328: 1829: 1796: 1736: 809: 698: 566: 427: 213: 210: 128: 331:
1. If the same is true for the integers, that is equivalent to the existence of a
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Barsotti–Tate representations. These are similar to finite flat representations.
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Reducible representations. These have a proper non-zero sub-representation.
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introduced a class of Galois representations of the absolute Galois group
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Absolutely irreducible representations. These remain irreducible over an
626: 79: 63: 20: 652: 132: 108: 1428: 1142:{\displaystyle r_{K}:C_{K}{\tilde {\rightarrow }}W_{K}^{\text{ab}}} 1021:
Unramified representations. These are trivial on the inertia group.
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Tamely ramified representations. These are trivial on the (first)
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Good representations. These are related to the representations of
685:. Artin's study of these representations led him to formulate the 979:
Modular representations. These are representations coming from a
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and the usual (Euclidean) topology on complex vector spaces, the
370:. This is an interesting question even (perhaps especially) when 268: 1486:
These representations are the same as the representations over
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Artin conductor § Artin representation and Artin character
1549:). These latter have the nice feature that the continuity of 1272:
whose image is infinite and therefore is not a character of
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has a normal integral basis. This may be seen by using the
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Automorphic forms, representations, and L-functions, Part 2
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with different meanings. Some of these conditions include:
549: 1705: 1308:, then the ℓ-adic cohomology of the geometric fibre of 78:-module. The study of Galois modules for extensions of 1557:, thus making the situation more algebraic in flavor. 1513:
sets up a bijection between ℓ-adic representations of
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representations over fields of positive characteristic
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and the ℓ-adic Tate modules of abelian varieties over
1088: 854: 527:. It is certainly therefore necessary for it to be a 1028: 546:
to embed the abelian field into a cyclotomic field.
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are Galois modules for the absolute Galois group of
1319:which, via φ, induces an ℓ-adic representation of 1141: 874: 1553:is only with respect to the discrete topology on 768:is either a finite-dimensional vector space over 289:be the corresponding Galois group. Then the ring 1841: 1781:, Fields Institute monographs, Providence, RI: 1344: 875:{\displaystyle \mathbf {Z} _{\ell }^{\times }} 1812:Galois module structure of algebraic integers 1715:Grundlehren der Mathematischen Wissenschaften 723:of an Artin representation is always finite. 674:finite-dimensional linear representations of 269:Galois module structure of algebraic integers 1709:; Schmidt, Alexander; Wingberg, Kay (2000), 1230:can have strictly more representations than 1667:)th power of the (arithmetic) Frobenius of 1567:Compatible system of ℓ-adic representations 1330:is a local field of residue characteristic 1717:, vol. 323, Berlin: Springer-Verlag, 1416:is a finite-dimensional vector space over 1296:. These arise naturally from geometry: if 1037:is a local or global field, the theory of 1216:can be considered as a representation of 848:under the ℓ-adic cyclotomic character is 726: 534:A classical result, based on a result of 1806: 1590: 1258:. Thus, the absolute value character on 902: 828:). The first examples to arise were the 636: 450:), as can be deduced from the theory of 74:, but can also be used as a synonym for 1750:(1979), "Number theoretic background", 550:Galois representations in number theory 1842: 1776: 1529:) and Weil–Deligne representations of 708:Because of the incompatibility of the 689:and conjecture what is now called the 184: 147:, its first cohomology group is zero). 1687: 1357:be a field of characteristic zero. A 1244:the continuous complex characters of 1746: 1643:is the size of the residue field of 976:Minimally ramified representations. 917:Local conditions on representations 13: 1770: 1289:is defined in the same way as for 1055:, a continuous group homomorphism 594:(see Hilbert–Speiser theorem). If 462:does not. This is an example of a 14: 1866: 1584: 1501:If the residue characteristic of 1279:(as all such have finite image). 1029:Representations of the Weil group 446:have normal integral bases (over 426:In fact all the subfields of the 281:be a Galois extension of a field 1388:a continuous group homomorphism 857: 246:. A Galois module ρ : 1312:is an ℓ-adic representation of 1251:are in bijection with those of 1209:. Via φ, any representation of 1596: 1572:Arboreal Galois representation 1118: 1115: 777:(the algebraic closure of the 1: 1783:American Mathematical Society 1681: 943:Crystabelline representations 909:Modular representation theory 1659:is equivalent to the − 1345:Weil–Deligne representations 1282:An ℓ-adic representation of 1018:Trianguline representations. 946:Crystalline representations. 895:they can be identified with 538:, is that a tamely ramified 54:is frequently used when the 7: 1711:Cohomology of Number Fields 1560: 1359:Weil–Deligne representation 971:Irreducible representations 967:Hodge–Tate representations. 830:ℓ-adic cyclotomic character 558:is a Galois extension of a 123:is a Galois module for the 97: 10: 1871: 1777:Snaith, Victor P. (1994), 1006:semistable elliptic curves 906: 640: 458:). On the other hand, the 1163:or the idele class group 608:global class field theory 1577: 1511:ℓ-adic monodromy theorem 1178:is local or global) and 983:, but can also refer to 949:de Rham representations. 604:local class field theory 586:for the Galois group of 579:is a Galois module over 474:). What matters here is 306:can be considered as an 197:(with valuation denoted 90:is an important tool in 16:Mathematical terminology 1850:Algebraic number theory 1779:Galois module structure 899:Artin representations. 602:and its study leads to 544:Kronecker–Weber theorem 456:Hilbert–Speiser theorem 275:algebraic number theory 1545:(or equivalently over 1353:be a local field. Let 1265:yields a character of 1174:(depending on whether 1143: 876: 727:ℓ-adic representations 472:perhaps known earlier? 352:give a free basis for 1505:is different from ℓ, 1205:of the Weil group of 1144: 907:Further information: 903:Mod ℓ representations 877: 737:ℓ-adic representation 687:Artin reciprocity law 683:complex vector spaces 668:Artin representations 641:Further information: 637:Artin representations 333:normal integral basis 125:absolute Galois group 72:representation theory 52:Galois representation 1086: 964:with good reduction. 852: 616:separable extensions 540:abelian number field 317:normal basis theorem 145:Hilbert's theorem 90 109:multiplicative group 1237:. For example, via 1138: 871: 651:be a number field. 627:ℓ-adic Tate modules 610:, the union of the 492:, and taking still 466:condition found by 185:Ramification theory 1808:Fröhlich, Albrecht 1492:Weil–Deligne group 1420:equipped with the 1302:projective variety 1139: 1124: 1078:topological groups 1013:ramification group 872: 855: 789:finitely generated 748:group homomorphism 710:profinite topology 612:idele class groups 481:. In terms of the 346:conjugate elements 300:algebraic integers 216:with Galois group 62:over a field or a 1763:978-0-8218-1437-6 1724:978-3-540-66671-4 1699:978-0-8218-1635-6 1422:discrete topology 1136: 1121: 935:algebraic closure 631:abelian varieties 622:is used instead. 521:projective module 428:cyclotomic fields 170:ℓ-adic cohomology 117:separable closure 1862: 1836: 1803: 1766: 1743: 1707:Neukirch, JĂŒrgen 1702: 1675: 1635: 1634: 1633: 1630: 1615: 1612: 1600: 1594: 1588: 1541: 1525: 1446: 1411: 1384:) consisting of 1200: 1199: 1198: 1195: 1187: 1184: 1148: 1146: 1145: 1140: 1137: 1134: 1132: 1123: 1122: 1114: 1111: 1110: 1098: 1097: 1071: 1039:class formations 887: 881: 879: 878: 873: 870: 865: 860: 824: 810:integral closure 804: 795: 773: 763: 746:is a continuous 691:Artin conjecture 670:. These are the 567:ring of integers 452:Gaussian periods 418: 399: 398: 385:For example, if 254:) is said to be 214:Galois extension 129:cohomology group 88:group cohomology 1870: 1869: 1865: 1864: 1863: 1861: 1860: 1859: 1840: 1839: 1826: 1816:Springer-Verlag 1793: 1773: 1771:Further reading 1764: 1725: 1700: 1690:Motives, Part 2 1684: 1679: 1678: 1672: 1655:) is such that 1641: 1631: 1626: 1617: 1616: 1613: 1608: 1607: 1606:|| is given by 1601: 1597: 1589: 1585: 1580: 1563: 1544: 1537: 1534: 1528: 1521: 1518: 1480: 1434: 1405: 1398: 1389: 1370: 1347: 1339: 1324: 1317: 1294: 1287: 1277: 1270: 1263: 1256: 1249: 1242: 1235: 1228: 1221: 1214: 1196: 1191: 1189: 1188: 1185: 1180: 1179: 1168: 1157: 1133: 1128: 1113: 1112: 1106: 1102: 1093: 1089: 1087: 1084: 1083: 1069: 1062: 1056: 1053: 1031: 962:elliptic curves 919: 911: 905: 890: 883: 866: 861: 856: 853: 850: 849: 847: 827: 820: 818: 807: 800: 799:-module (where 798: 791: 786: 776: 769: 756: 750: 744: 729: 717: 693:concerning the 679: 660: 645: 639: 584: 573: 552: 517: 416: 396: 394: 376:rational number 369: 360: 343: 335:, i.e. of α in 319:one knows that 314: 297: 271: 263: 240: 187: 174:geometric fibre 100: 17: 12: 11: 5: 1868: 1858: 1857: 1852: 1838: 1837: 1824: 1804: 1791: 1772: 1769: 1768: 1767: 1762: 1744: 1723: 1703: 1698: 1683: 1680: 1677: 1676: 1670: 1639: 1595: 1582: 1581: 1579: 1576: 1575: 1574: 1569: 1562: 1559: 1542: 1532: 1526: 1516: 1484: 1483: 1478: 1425: 1401: 1396: 1372:(or simply of 1368: 1346: 1343: 1337: 1322: 1315: 1292: 1285: 1275: 1268: 1261: 1254: 1247: 1240: 1233: 1226: 1219: 1212: 1203:abelianization 1166: 1155: 1150: 1149: 1131: 1127: 1120: 1117: 1109: 1105: 1101: 1096: 1092: 1067: 1060: 1051: 1030: 1027: 1026: 1025: 1022: 1019: 1016: 1009: 1002: 999: 992: 988: 977: 974: 968: 965: 958: 950: 947: 944: 941: 938: 931: 918: 915: 904: 901: 888: 869: 864: 859: 843: 825: 816: 805: 796: 784: 779:ℓ-adic numbers 774: 754: 742: 728: 725: 715: 677: 658: 638: 635: 614:of all finite 582: 571: 551: 548: 515: 460:Gaussian field 436:roots of unity 424: 423: 365: 356: 344:such that its 339: 310: 293: 270: 267: 261: 238: 186: 183: 182: 181: 172:groups of its 148: 103:Given a field 99: 96: 15: 9: 6: 4: 3: 2: 1867: 1856: 1855:Galois theory 1853: 1851: 1848: 1847: 1845: 1835: 1831: 1827: 1825:3-540-11920-5 1821: 1817: 1813: 1809: 1805: 1802: 1798: 1794: 1792:0-8218-0264-X 1788: 1784: 1780: 1775: 1774: 1765: 1759: 1755: 1754: 1749: 1745: 1742: 1738: 1734: 1730: 1726: 1720: 1716: 1712: 1708: 1704: 1701: 1695: 1691: 1686: 1685: 1673: 1666: 1662: 1658: 1654: 1650: 1646: 1642: 1629: 1624: 1620: 1611: 1605: 1599: 1592: 1591:Fröhlich 1983 1587: 1583: 1573: 1570: 1568: 1565: 1564: 1558: 1556: 1552: 1548: 1540: 1535: 1524: 1519: 1512: 1508: 1504: 1499: 1497: 1493: 1489: 1481: 1475: âˆˆ  1474: 1470: 1466: 1462: 1458: 1454: 1450: 1445: 1441: 1437: 1433: 1430: 1426: 1423: 1419: 1415: 1409: 1404: 1399: 1392: 1387: 1386: 1385: 1383: 1379: 1376:) is a pair ( 1375: 1371: 1364: 1360: 1356: 1352: 1342: 1340: 1333: 1329: 1325: 1318: 1311: 1307: 1303: 1299: 1295: 1288: 1280: 1278: 1271: 1264: 1257: 1250: 1243: 1236: 1229: 1222: 1215: 1208: 1204: 1194: 1183: 1177: 1173: 1169: 1162: 1158: 1129: 1125: 1107: 1103: 1099: 1094: 1090: 1082: 1081: 1080: 1079: 1075: 1070: 1063: 1054: 1048: 1044: 1040: 1036: 1023: 1020: 1017: 1014: 1010: 1007: 1003: 1000: 997: 993: 989: 986: 982: 978: 975: 972: 969: 966: 963: 959: 956: 951: 948: 945: 942: 939: 937:of the field. 936: 932: 929: 925: 924: 923: 914: 910: 900: 898: 894: 886: 867: 862: 846: 842: 837: 835: 831: 823: 815: 811: 803: 794: 790: 783: 780: 772: 767: 761: 757: 749: 745: 738: 734: 724: 722: 718: 711: 706: 704: 702: 696: 692: 688: 684: 680: 673: 669: 666:, now called 665: 661: 654: 650: 644: 634: 632: 628: 623: 621: 617: 613: 609: 605: 601: 597: 593: 589: 585: 578: 574: 568: 564: 561: 557: 547: 545: 541: 537: 536:David Hilbert 532: 530: 526: 522: 518: 511: 508:to the power 507: 503: 499: 496: =  495: 491: 487: 484: 480: 477: 473: 469: 465: 461: 457: 453: 449: 445: 441: 437: 433: 429: 421: 414: 411: 410: 409: 407: 403: 392: 389: =  388: 383: 381: 377: 373: 368: 364: 359: 355: 351: 347: 342: 338: 334: 330: 326: 322: 318: 313: 309: 305: 301: 296: 292: 288: 284: 280: 276: 273:In classical 266: 264: 257: 253: 249: 245: 244:inertia group 241: 234: 230: 226: 223: 219: 215: 212: 208: 204: 200: 196: 192: 179: 175: 171: 167: 164:over a field 163: 160: 157: 153: 149: 146: 142: 138: 134: 130: 127:. Its second 126: 122: 118: 114: 110: 106: 102: 101: 95: 93: 92:number theory 89: 85: 84:global fields 81: 77: 73: 69: 65: 61: 58:-module is a 57: 53: 49: 45: 41: 37: 33: 31: 26: 25:Galois module 22: 1811: 1778: 1752: 1714: 1710: 1689: 1668: 1664: 1660: 1656: 1652: 1648: 1644: 1637: 1627: 1622: 1618: 1609: 1603: 1598: 1586: 1554: 1550: 1546: 1538: 1530: 1522: 1514: 1507:Grothendieck 1502: 1500: 1495: 1487: 1485: 1476: 1472: 1468: 1464: 1460: 1456: 1452: 1448: 1443: 1439: 1435: 1432:endomorphism 1417: 1413: 1407: 1402: 1394: 1390: 1381: 1377: 1373: 1366: 1362: 1358: 1354: 1350: 1348: 1335: 1331: 1327: 1320: 1313: 1309: 1305: 1300:is a smooth 1297: 1290: 1283: 1281: 1273: 1266: 1259: 1252: 1245: 1238: 1231: 1224: 1217: 1210: 1206: 1192: 1181: 1175: 1171: 1164: 1160: 1153: 1151: 1065: 1058: 1049: 1042: 1041:attaches to 1034: 1032: 995: 994:Potentially 981:modular form 955:group scheme 920: 912: 896: 892: 884: 844: 840: 838: 833: 821: 813: 801: 792: 781: 770: 765: 759: 752: 740: 736: 733:prime number 730: 713: 707: 700: 675: 667: 663: 656: 648: 646: 624: 619: 599: 595: 591: 587: 580: 576: 569: 562: 560:number field 555: 553: 533: 528: 524: 513: 509: 505: 504:must divide 501: 497: 493: 489: 485: 483:discriminant 479:ramification 475: 471: 468:Emmy Noether 463: 447: 444:prime number 443: 439: 431: 425: 419: 412: 405: 401: 390: 386: 384: 379: 371: 366: 362: 357: 353: 349: 340: 336: 332: 324: 320: 311: 307: 303: 294: 290: 286: 282: 278: 272: 259: 255: 251: 247: 236: 232: 228: 224: 217: 206: 202: 198: 195:valued field 190: 188: 177: 165: 151: 140: 137:Brauer group 120: 112: 104: 75: 60:vector space 55: 51: 40:Galois group 35: 29: 24: 18: 1593:, p. 8 1223:. However, 1074:isomorphism 731:Let ℓ be a 500:, no prime 327:-module of 242:denote its 64:free module 50:. The term 21:mathematics 1844:Categories 1834:0501.12012 1801:0830.11042 1748:Tate, John 1741:0948.11001 1682:References 1463:)= || 1447:such that 1047:Weil group 703:-functions 695:holomorphy 672:continuous 653:Emil Artin 323:is a free 285:, and let 258:if ρ( 256:unramified 201:) and let 133:isomorphic 86:and their 38:being the 1429:nilpotent 1119:~ 1116:→ 1072:, and an 1057:φ : 996:something 897:bona fide 868:× 863:ℓ 751:ρ : 464:necessary 265:) = {1}. 222:extension 220:. For an 168:then the 44:extension 1810:(1983), 1561:See also 1471:for all 1438: : 1412:, where 1393: : 519:to be a 408:) where 98:Examples 42:of some 1733:1737196 1602:Here || 1490:of the 1380:,  1201:is the 928:abelian 808:is the 787:) or a 415:= exp(2 395:√ 374:is the 135:to the 115:) of a 66:over a 34:, with 32:-module 1832:  1822:  1799:  1789:  1760:  1739:  1731:  1721:  1696:  1636:where 1632:  1614:  1520:(over 1197:  1186:  1152:where 764:where 758:→ Aut( 699:Artin 606:. For 565:, the 378:field 348:under 277:, let 250:→ Aut( 235:, let 211:finite 162:scheme 159:proper 156:smooth 107:, the 48:fields 1578:Notes 1536:over 1400:→ Aut 1361:over 1326:. If 1304:over 891:with 735:. An 721:image 523:over 454:(the 361:over 209:be a 193:be a 154:is a 80:local 27:is a 1820:ISBN 1787:ISBN 1758:ISBN 1719:ISBN 1694:ISBN 1647:and 1349:Let 1045:its 647:Let 529:free 476:tame 438:for 434:-th 430:for 422:/3). 329:rank 189:Let 143:(by 68:ring 23:, a 1830:Zbl 1797:Zbl 1737:Zbl 1509:'s 1494:of 1365:of 1159:is 1076:of 1033:If 819:in 812:of 739:of 712:on 697:of 681:on 662:of 629:of 618:of 575:of 488:of 302:of 298:of 231:to 227:of 150:If 139:of 131:is 119:of 82:or 70:in 46:of 19:In 1846:: 1828:, 1818:, 1795:, 1785:, 1735:, 1729:MR 1727:, 1713:, 1498:. 1467:|| 1455:)N 1442:→ 1427:a 1341:. 1190:ab 1135:ab 1064:→ 705:. 633:. 442:a 397:−3 382:. 94:. 1674:. 1671:K 1669:W 1665:w 1663:( 1661:v 1657:w 1653:w 1651:( 1649:v 1645:K 1640:K 1638:q 1628:K 1625:) 1623:w 1621:( 1619:v 1610:q 1604:w 1555:V 1551:r 1547:C 1543:ℓ 1539:Q 1533:K 1531:W 1527:ℓ 1523:Q 1517:K 1515:W 1503:K 1496:K 1488:E 1482:. 1479:K 1477:W 1473:w 1469:N 1465:w 1461:w 1459:( 1457:r 1453:w 1451:( 1449:r 1444:V 1440:V 1436:N 1424:, 1418:E 1414:V 1410:) 1408:V 1406:( 1403:E 1397:K 1395:W 1391:r 1382:N 1378:r 1374:K 1369:K 1367:W 1363:E 1355:E 1351:K 1338:K 1336:W 1332:p 1328:K 1323:K 1321:W 1316:K 1314:G 1310:X 1306:K 1298:X 1293:K 1291:G 1286:K 1284:W 1276:K 1274:G 1269:K 1267:W 1262:K 1260:C 1255:K 1253:C 1248:K 1246:W 1241:K 1239:r 1234:K 1232:G 1227:K 1225:W 1220:K 1218:W 1213:K 1211:G 1207:K 1193:K 1182:W 1176:K 1172:K 1170:/ 1167:K 1165:I 1161:K 1156:K 1154:C 1130:K 1126:W 1108:K 1104:C 1100:: 1095:K 1091:r 1068:K 1066:G 1061:K 1059:W 1052:K 1050:W 1043:K 1035:K 1015:. 1008:. 987:. 957:. 930:. 893:C 889:ℓ 885:Q 858:Z 845:Q 841:G 834:K 826:ℓ 822:Q 817:ℓ 814:Z 806:ℓ 802:Z 797:ℓ 793:Z 785:ℓ 782:Q 775:ℓ 771:Q 766:M 762:) 760:M 755:K 753:G 743:K 741:G 716:K 714:G 701:L 678:K 676:G 664:K 659:K 657:G 649:K 620:K 600:K 596:K 592:K 590:/ 588:L 583:K 581:O 577:L 572:L 570:O 563:K 556:L 525:Z 516:L 514:O 510:p 506:D 502:p 498:Q 494:K 490:L 486:D 470:( 448:Z 440:p 432:p 420:i 417:π 413:ζ 406:ζ 404:( 402:Q 393:( 391:Q 387:L 380:Q 372:K 367:K 363:O 358:L 354:O 350:G 341:L 337:O 325:K 321:L 312:K 308:O 304:L 295:L 291:O 287:G 283:K 279:L 262:w 260:I 252:V 248:G 239:w 237:I 233:L 229:v 225:w 218:G 207:K 205:/ 203:L 199:v 191:K 180:. 178:K 166:K 152:X 141:K 121:K 113:K 111:( 105:K 76:G 56:G 36:G 30:G

Index

mathematics
G-module
Galois group
extension
fields
vector space
free module
ring
representation theory
local
global fields
group cohomology
number theory
multiplicative group
separable closure
absolute Galois group
cohomology group
isomorphic
Brauer group
Hilbert's theorem 90
smooth
proper
scheme
ℓ-adic cohomology
geometric fibre
valued field
finite
Galois extension
extension
inertia group

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