2195:
for each row of the matrix, and by using the matrix coefficients within each column to assign each matroid element a linear combination of these transcendentals. The converse is false: not every algebraic matroid has a linear representation.
1891:
1801:
246:
1187:
538:
1698:
1582:
1082:
1257:
1290:
491:
454:
1012:
1913:
1831:
1755:
1720:
1641:
1619:
1545:
1499:
1381:
1139:
200:
1117:
807:
1325:
1462:
1519:
1041:
849:
1407:
2316:
1435:
1353:
1958:
2185:
2146:
2117:
2097:
2073:
2053:
2033:
2002:
1982:
1664:
977:
957:
933:
913:
893:
873:
827:
778:
751:
731:
704:
681:
643:
2337:
2252:
2279:
2224:
100:
2187:, in which the matroid elements correspond to matrix columns, and a set of elements is independent if the corresponding set of columns is
1732:
2383:
636:
588:
2152:. No good characterization of algebraic matroids is known, but certain matroids are known to be non-algebraic; the smallest is the
1836:
2311:
2191:. Every matroid with a linear representation of this type may also be represented as an algebraic matroid, by choosing an
1410:
629:
346:
1760:
106:
1361:
are known to be transcendental, it is not known whether the set of both of them is algebraically independent over
210:
2429:
1356:
1147:
2192:
581:
505:
384:
334:
2079:. In this matroid, the rank of a set of elements is its transcendence degree, and the flat generated by a set
2434:
393:
86:
2160:
852:
550:
401:
352:
133:
1669:
1553:
1057:
1044:
754:
1199:
1265:
467:
430:
2164:
2009:
1808:
274:
148:
993:
2188:
711:
684:
556:
364:
315:
260:
154:
140:
68:
36:
1896:
1814:
1738:
1703:
1624:
1587:
1528:
1467:
1364:
1122:
183:
1087:
786:
569:
127:
55:
1295:
2405:
2379:
2373:
2285:
2275:
2220:
1924:
1440:
610:
407:
172:
113:
1504:
1017:
834:
2346:
2293:
2153:
1804:
1386:
657:
616:
602:
416:
358:
321:
121:
94:
80:
2358:
1964:
can be used to show that there always exists a maximal algebraically independent subset of
2354:
2297:
1930:
1420:
1338:
1048:
936:
378:
328:
166:
1961:
1935:
496:
17:
2170:
2122:
2102:
2082:
2058:
2038:
2018:
1987:
1967:
1649:
1522:
1193:
algebraically independent over the rational numbers, because the nontrivial polynomial
962:
942:
918:
898:
878:
858:
829:
812:
763:
736:
716:
689:
666:
422:
2274:. Encyclopaedia of Mathematical Sciences. Vol. 49 (Second ed.). p. 61.
2423:
2267:
1052:
563:
459:
74:
2247:
595:
370:
266:
2250:(2008), "III.41 Irrational and Transcendental Numbers", in Gowers, Timothy (ed.),
2214:
2005:
988:
575:
286:
160:
42:
757:
340:
2289:
1735:
can often be used to prove that some sets are algebraically independent over
1047:: they are not the roots of any nontrivial polynomial whose coefficients are
2410:
300:
205:
2350:
2004:. Further, all the maximal algebraically independent subsets have the same
294:
280:
2076:
178:
62:
2335:
Ingleton, A. W.; Main, R. A. (1975), "Non-algebraic matroids exist",
661:
875:. In general, all the elements of an algebraically independent set
2148:. A matroid that can be generated in this way is called an
2075:
satisfy the axioms that define the independent sets of a
2314:(1996). "Modular Functions and Transcendence Problems".
1886:{\displaystyle e^{\alpha _{1}},\ldots ,e^{\alpha _{n}}}
2173:
2125:
2105:
2085:
2061:
2041:
2021:
1990:
1970:
1938:
1899:
1839:
1817:
1763:
1741:
1706:
1672:
1652:
1627:
1590:
1556:
1531:
1507:
1470:
1443:
1423:
1389:
1367:
1341:
1298:
1268:
1202:
1150:
1125:
1090:
1060:
1020:
996:
965:
945:
921:
901:
881:
861:
837:
815:
789:
766:
739:
719:
692:
669:
508:
470:
433:
213:
186:
2317:Comptes Rendus de l'Académie des Sciences, Série I
2179:
2140:
2111:
2091:
2067:
2047:
2027:
1996:
1976:
1952:
1907:
1885:
1825:
1795:
1749:
1714:
1692:
1658:
1635:
1613:
1576:
1539:
1513:
1493:
1456:
1429:
1401:
1375:
1347:
1319:
1284:
1251:
1181:
1133:
1111:
1076:
1035:
1006:
971:
951:
927:
907:
887:
867:
843:
821:
801:
772:
745:
725:
698:
675:
532:
485:
448:
240:
194:
1796:{\displaystyle \alpha _{1},\ldots ,\alpha _{n}}
637:
8:
1176:
1151:
1119:is algebraically independent over the field
1106:
1091:
1071:
1061:
796:
790:
241:{\displaystyle 0=\mathbb {Z} /1\mathbb {Z} }
27:Set without nontrivial polynomial equalities
2338:Bulletin of the London Mathematical Society
2055:, the algebraically independent subsets of
1182:{\displaystyle \{{\sqrt {\pi }},2\pi +1\}}
644:
630:
31:
2256:, Princeton University Press, p. 222
2172:
2124:
2104:
2084:
2060:
2040:
2020:
1989:
1969:
1942:
1937:
1901:
1900:
1898:
1875:
1870:
1849:
1844:
1838:
1819:
1818:
1816:
1787:
1768:
1762:
1743:
1742:
1740:
1708:
1707:
1705:
1681:
1677:
1671:
1651:
1629:
1628:
1626:
1600:
1589:
1565:
1561:
1555:
1533:
1532:
1530:
1506:
1480:
1469:
1448:
1442:
1422:
1388:
1369:
1368:
1366:
1340:
1331:Algebraic independence of known constants
1297:
1275:
1267:
1231:
1201:
1154:
1149:
1127:
1126:
1124:
1089:
1064:
1059:
1019:
997:
995:
964:
944:
920:
900:
880:
860:
836:
814:
788:
765:
738:
718:
691:
668:
533:{\displaystyle \mathbb {Z} (p^{\infty })}
521:
510:
509:
507:
477:
473:
472:
469:
440:
436:
435:
432:
234:
233:
225:
221:
220:
212:
188:
187:
185:
1893:are also algebraically independent over
2205:
959:generated by the remaining elements of
34:
2378:, New Age International, p. 909,
2253:The Princeton Companion to Mathematics
1525:, are algebraically independent over
915:are by necessity transcendental over
7:
2272:Introduction to Modern Number Theory
101:Free product of associative algebras
2099:of elements is the intersection of
1693:{\displaystyle e^{\pi {\sqrt {n}}}}
1621:are algebraically independent over
1577:{\displaystyle e^{\pi {\sqrt {3}}}}
1383:. In fact, it is not even known if
1700:is algebraically independent over
1591:
1508:
1471:
1077:{\displaystyle \{{\sqrt {\pi }}\}}
809:is algebraically independent over
522:
25:
1252:{\displaystyle P(x,y)=2x^{2}-y+1}
783:In particular, a one element set
589:Noncommutative algebraic geometry
1285:{\displaystyle x={\sqrt {\pi }}}
486:{\displaystyle \mathbb {Q} _{p}}
449:{\displaystyle \mathbb {Z} _{p}}
2135:
2129:
1608:
1594:
1488:
1474:
1218:
1206:
1007:{\displaystyle {\sqrt {\pi }}}
760:equation with coefficients in
527:
514:
1:
2270:; Panchishkin, A. A. (2007).
1733:Lindemann–Weierstrass theorem
1727:Lindemann–Weierstrass theorem
2159:Many finite matroids may be
1908:{\displaystyle \mathbb {Q} }
1826:{\displaystyle \mathbb {Q} }
1750:{\displaystyle \mathbb {Q} }
1715:{\displaystyle \mathbb {Q} }
1636:{\displaystyle \mathbb {Q} }
1614:{\displaystyle \Gamma (1/3)}
1540:{\displaystyle \mathbb {Q} }
1494:{\displaystyle \Gamma (1/4)}
1376:{\displaystyle \mathbb {Q} }
1134:{\displaystyle \mathbb {Q} }
195:{\displaystyle \mathbb {Z} }
2406:"Algebraically Independent"
2375:Applied Discrete Structures
1112:{\displaystyle \{2\pi +1\}}
802:{\displaystyle \{\alpha \}}
347:Unique factorization domain
2451:
1922:
1757:. It states that whenever
1646:for all positive integers
107:Tensor product of algebras
2219:. Springer. p. 174.
1320:{\displaystyle y=2\pi +1}
708:algebraically independent
18:Algebraically independent
2213:Patrick Morandi (1996).
1457:{\displaystyle e^{\pi }}
1051:. Thus, each of the two
385:Formal power series ring
335:Integrally closed domain
2216:Field and Galois Theory
1960:that is not algebraic,
1514:{\displaystyle \Gamma }
1036:{\displaystyle 2\pi +1}
844:{\displaystyle \alpha }
753:do not satisfy any non-
394:Algebraic number theory
87:Total ring of fractions
2181:
2142:
2113:
2093:
2069:
2049:
2029:
1998:
1978:
1954:
1909:
1887:
1827:
1797:
1751:
1716:
1694:
1660:
1637:
1615:
1578:
1541:
1515:
1495:
1458:
1431:
1403:
1402:{\displaystyle \pi +e}
1377:
1349:
1321:
1286:
1253:
1183:
1135:
1113:
1078:
1045:transcendental numbers
1037:
1008:
973:
953:
935:, and over all of the
929:
909:
889:
869:
845:
823:
803:
774:
747:
727:
700:
677:
551:Noncommutative algebra
534:
487:
450:
402:Algebraic number field
353:Principal ideal domain
242:
196:
134:Frobenius endomorphism
2372:Joshi, K. D. (1997),
2182:
2143:
2114:
2094:
2070:
2050:
2030:
1999:
1979:
1955:
1910:
1888:
1828:
1798:
1752:
1717:
1695:
1661:
1638:
1616:
1579:
1542:
1516:
1496:
1459:
1432:
1413:proved in 1996 that:
1404:
1378:
1350:
1322:
1287:
1254:
1184:
1141:of rational numbers.
1136:
1114:
1079:
1038:
1009:
974:
954:
930:
910:
890:
870:
846:
824:
804:
775:
748:
728:
701:
678:
535:
488:
451:
243:
197:
2351:10.1112/blms/7.2.144
2189:linearly independent
2171:
2123:
2103:
2083:
2059:
2039:
2019:
2010:transcendence degree
1988:
1968:
1936:
1897:
1837:
1815:
1809:linearly independent
1761:
1739:
1704:
1670:
1650:
1625:
1588:
1554:
1529:
1505:
1468:
1441:
1430:{\displaystyle \pi }
1421:
1387:
1365:
1348:{\displaystyle \pi }
1339:
1296:
1266:
1200:
1148:
1123:
1088:
1058:
1018:
994:
963:
943:
919:
899:
879:
859:
835:
813:
787:
764:
737:
717:
690:
667:
557:Noncommutative rings
506:
468:
431:
275:Non-associative ring
211:
184:
141:Algebraic structures
1953:{\displaystyle L/K}
733:if the elements of
316:Commutative algebra
155:Associative algebra
37:Algebraic structure
2312:Nesterenko, Yuri V
2177:
2138:
2109:
2089:
2065:
2045:
2025:
2012:of the extension.
1994:
1974:
1950:
1919:Algebraic matroids
1905:
1883:
1823:
1793:
1747:
1712:
1690:
1656:
1633:
1611:
1574:
1537:
1511:
1491:
1454:
1427:
1399:
1373:
1345:
1317:
1282:
1249:
1179:
1131:
1109:
1074:
1033:
1004:
969:
949:
925:
905:
885:
865:
841:
819:
799:
770:
743:
723:
696:
673:
570:Semiprimitive ring
530:
483:
446:
254:Related structures
238:
192:
128:Inner automorphism
114:Ring homomorphisms
2281:978-3-540-20364-3
2226:978-0-387-94753-2
2180:{\displaystyle K}
2150:algebraic matroid
2141:{\displaystyle K}
2112:{\displaystyle L}
2092:{\displaystyle T}
2068:{\displaystyle S}
2048:{\displaystyle L}
2028:{\displaystyle S}
1997:{\displaystyle K}
1977:{\displaystyle L}
1925:Algebraic matroid
1805:algebraic numbers
1686:
1659:{\displaystyle n}
1570:
1280:
1159:
1144:However, the set
1069:
1002:
972:{\displaystyle S}
952:{\displaystyle K}
928:{\displaystyle K}
908:{\displaystyle K}
888:{\displaystyle S}
868:{\displaystyle K}
822:{\displaystyle K}
773:{\displaystyle K}
746:{\displaystyle S}
726:{\displaystyle K}
699:{\displaystyle L}
676:{\displaystyle S}
654:
653:
611:Geometric algebra
322:Commutative rings
173:Category of rings
16:(Redirected from
2442:
2430:Abstract algebra
2416:
2415:
2390:
2388:
2369:
2363:
2361:
2332:
2326:
2325:
2308:
2302:
2301:
2264:
2258:
2257:
2244:
2238:
2237:
2235:
2233:
2210:
2186:
2184:
2183:
2178:
2147:
2145:
2144:
2139:
2118:
2116:
2115:
2110:
2098:
2096:
2095:
2090:
2074:
2072:
2071:
2066:
2054:
2052:
2051:
2046:
2034:
2032:
2031:
2026:
2003:
2001:
2000:
1995:
1983:
1981:
1980:
1975:
1959:
1957:
1956:
1951:
1946:
1914:
1912:
1911:
1906:
1904:
1892:
1890:
1889:
1884:
1882:
1881:
1880:
1879:
1856:
1855:
1854:
1853:
1832:
1830:
1829:
1824:
1822:
1802:
1800:
1799:
1794:
1792:
1791:
1773:
1772:
1756:
1754:
1753:
1748:
1746:
1721:
1719:
1718:
1713:
1711:
1699:
1697:
1696:
1691:
1689:
1688:
1687:
1682:
1665:
1663:
1662:
1657:
1642:
1640:
1639:
1634:
1632:
1620:
1618:
1617:
1612:
1604:
1583:
1581:
1580:
1575:
1573:
1572:
1571:
1566:
1546:
1544:
1543:
1538:
1536:
1520:
1518:
1517:
1512:
1500:
1498:
1497:
1492:
1484:
1463:
1461:
1460:
1455:
1453:
1452:
1436:
1434:
1433:
1428:
1408:
1406:
1405:
1400:
1382:
1380:
1379:
1374:
1372:
1354:
1352:
1351:
1346:
1326:
1324:
1323:
1318:
1291:
1289:
1288:
1283:
1281:
1276:
1258:
1256:
1255:
1250:
1236:
1235:
1188:
1186:
1185:
1180:
1160:
1155:
1140:
1138:
1137:
1132:
1130:
1118:
1116:
1115:
1110:
1083:
1081:
1080:
1075:
1070:
1065:
1049:rational numbers
1042:
1040:
1039:
1034:
1013:
1011:
1010:
1005:
1003:
998:
978:
976:
975:
970:
958:
956:
955:
950:
937:field extensions
934:
932:
931:
926:
914:
912:
911:
906:
894:
892:
891:
886:
874:
872:
871:
866:
850:
848:
847:
842:
828:
826:
825:
820:
808:
806:
805:
800:
779:
777:
776:
771:
752:
750:
749:
744:
732:
730:
729:
724:
705:
703:
702:
697:
682:
680:
679:
674:
658:abstract algebra
646:
639:
632:
617:Operator algebra
603:Clifford algebra
539:
537:
536:
531:
526:
525:
513:
492:
490:
489:
484:
482:
481:
476:
455:
453:
452:
447:
445:
444:
439:
417:Ring of integers
411:
408:Integers modulo
359:Euclidean domain
247:
245:
244:
239:
237:
229:
224:
201:
199:
198:
193:
191:
95:Product of rings
81:Fractional ideal
40:
32:
21:
2450:
2449:
2445:
2444:
2443:
2441:
2440:
2439:
2420:
2419:
2403:
2402:
2399:
2394:
2393:
2386:
2371:
2370:
2366:
2334:
2333:
2329:
2310:
2309:
2305:
2282:
2266:
2265:
2261:
2246:
2245:
2241:
2231:
2229:
2227:
2212:
2211:
2207:
2202:
2169:
2168:
2121:
2120:
2119:with the field
2101:
2100:
2081:
2080:
2057:
2056:
2037:
2036:
2035:of elements of
2017:
2016:
2008:, known as the
1986:
1985:
1966:
1965:
1934:
1933:
1931:field extension
1927:
1921:
1895:
1894:
1871:
1866:
1845:
1840:
1835:
1834:
1813:
1812:
1783:
1764:
1759:
1758:
1737:
1736:
1729:
1702:
1701:
1673:
1668:
1667:
1648:
1647:
1623:
1622:
1586:
1585:
1557:
1552:
1551:
1527:
1526:
1503:
1502:
1466:
1465:
1444:
1439:
1438:
1419:
1418:
1409:is irrational.
1385:
1384:
1363:
1362:
1337:
1336:
1333:
1294:
1293:
1264:
1263:
1227:
1198:
1197:
1146:
1145:
1121:
1120:
1086:
1085:
1056:
1055:
1016:
1015:
992:
991:
985:
961:
960:
941:
940:
917:
916:
897:
896:
877:
876:
857:
856:
833:
832:
811:
810:
785:
784:
762:
761:
735:
734:
715:
714:
688:
687:
665:
664:
650:
621:
620:
553:
543:
542:
517:
504:
503:
471:
466:
465:
434:
429:
428:
409:
379:Polynomial ring
329:Integral domain
318:
308:
307:
209:
208:
182:
181:
167:Involutive ring
52:
41:
35:
28:
23:
22:
15:
12:
11:
5:
2448:
2446:
2438:
2437:
2435:Matroid theory
2432:
2422:
2421:
2418:
2417:
2404:Chen, Johnny.
2398:
2397:External links
2395:
2392:
2391:
2384:
2364:
2345:(2): 144–146,
2327:
2324:(10): 909–914.
2303:
2280:
2259:
2239:
2225:
2204:
2203:
2201:
2198:
2176:
2137:
2134:
2131:
2128:
2108:
2088:
2064:
2044:
2024:
2015:For every set
1993:
1973:
1949:
1945:
1941:
1923:Main article:
1920:
1917:
1903:
1878:
1874:
1869:
1865:
1862:
1859:
1852:
1848:
1843:
1821:
1790:
1786:
1782:
1779:
1776:
1771:
1767:
1745:
1728:
1725:
1724:
1723:
1710:
1685:
1680:
1676:
1655:
1644:
1631:
1610:
1607:
1603:
1599:
1596:
1593:
1569:
1564:
1560:
1548:
1535:
1523:gamma function
1510:
1490:
1487:
1483:
1479:
1476:
1473:
1451:
1447:
1426:
1398:
1395:
1392:
1371:
1344:
1335:Although both
1332:
1329:
1316:
1313:
1310:
1307:
1304:
1301:
1279:
1274:
1271:
1260:
1259:
1248:
1245:
1242:
1239:
1234:
1230:
1226:
1223:
1220:
1217:
1214:
1211:
1208:
1205:
1178:
1175:
1172:
1169:
1166:
1163:
1158:
1153:
1129:
1108:
1105:
1102:
1099:
1096:
1093:
1073:
1068:
1063:
1053:singleton sets
1032:
1029:
1026:
1023:
1001:
984:
981:
968:
948:
924:
904:
884:
864:
853:transcendental
840:
830:if and only if
818:
798:
795:
792:
769:
742:
722:
695:
672:
652:
651:
649:
648:
641:
634:
626:
623:
622:
614:
613:
585:
584:
578:
572:
566:
554:
549:
548:
545:
544:
541:
540:
529:
524:
520:
516:
512:
493:
480:
475:
456:
443:
438:
426:-adic integers
419:
413:
404:
390:
389:
388:
387:
381:
375:
374:
373:
361:
355:
349:
343:
337:
319:
314:
313:
310:
309:
306:
305:
304:
303:
291:
290:
289:
283:
271:
270:
269:
251:
250:
249:
248:
236:
232:
228:
223:
219:
216:
202:
190:
169:
163:
157:
151:
137:
136:
130:
124:
110:
109:
103:
97:
91:
90:
89:
83:
71:
65:
53:
51:Basic concepts
50:
49:
46:
45:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2447:
2436:
2433:
2431:
2428:
2427:
2425:
2413:
2412:
2407:
2401:
2400:
2396:
2387:
2385:9788122408263
2381:
2377:
2376:
2368:
2365:
2360:
2356:
2352:
2348:
2344:
2340:
2339:
2331:
2328:
2323:
2319:
2318:
2313:
2307:
2304:
2299:
2295:
2291:
2287:
2283:
2277:
2273:
2269:
2268:Manin, Yu. I.
2263:
2260:
2255:
2254:
2249:
2243:
2240:
2228:
2222:
2218:
2217:
2209:
2206:
2199:
2197:
2194:
2193:indeterminate
2190:
2174:
2167:over a field
2166:
2162:
2157:
2155:
2154:Vámos matroid
2151:
2132:
2126:
2106:
2086:
2078:
2062:
2042:
2022:
2013:
2011:
2007:
1991:
1971:
1963:
1947:
1943:
1939:
1932:
1926:
1918:
1916:
1876:
1872:
1867:
1863:
1860:
1857:
1850:
1846:
1841:
1810:
1806:
1788:
1784:
1780:
1777:
1774:
1769:
1765:
1734:
1726:
1683:
1678:
1674:
1666:, the number
1653:
1645:
1605:
1601:
1597:
1567:
1562:
1558:
1549:
1524:
1485:
1481:
1477:
1449:
1445:
1424:
1416:
1415:
1414:
1412:
1396:
1393:
1390:
1360:
1359:
1342:
1330:
1328:
1314:
1311:
1308:
1305:
1302:
1299:
1277:
1272:
1269:
1262:is zero when
1246:
1243:
1240:
1237:
1232:
1228:
1224:
1221:
1215:
1212:
1209:
1203:
1196:
1195:
1194:
1192:
1173:
1170:
1167:
1164:
1161:
1156:
1142:
1103:
1100:
1097:
1094:
1066:
1054:
1050:
1046:
1030:
1027:
1024:
1021:
999:
990:
982:
980:
966:
946:
938:
922:
902:
882:
862:
854:
838:
831:
816:
793:
781:
767:
759:
756:
740:
720:
713:
709:
693:
686:
670:
663:
659:
647:
642:
640:
635:
633:
628:
627:
625:
624:
619:
618:
612:
608:
607:
606:
605:
604:
599:
598:
597:
592:
591:
590:
583:
579:
577:
573:
571:
567:
565:
564:Division ring
561:
560:
559:
558:
552:
547:
546:
518:
502:
500:
494:
478:
464:
463:-adic numbers
462:
457:
441:
427:
425:
420:
418:
414:
412:
405:
403:
399:
398:
397:
396:
395:
386:
382:
380:
376:
372:
368:
367:
366:
362:
360:
356:
354:
350:
348:
344:
342:
338:
336:
332:
331:
330:
326:
325:
324:
323:
317:
312:
311:
302:
298:
297:
296:
292:
288:
284:
282:
278:
277:
276:
272:
268:
264:
263:
262:
258:
257:
256:
255:
230:
226:
217:
214:
207:
206:Terminal ring
203:
180:
176:
175:
174:
170:
168:
164:
162:
158:
156:
152:
150:
146:
145:
144:
143:
142:
135:
131:
129:
125:
123:
119:
118:
117:
116:
115:
108:
104:
102:
98:
96:
92:
88:
84:
82:
78:
77:
76:
75:Quotient ring
72:
70:
66:
64:
60:
59:
58:
57:
48:
47:
44:
39:→ Ring theory
38:
33:
30:
19:
2409:
2374:
2367:
2342:
2336:
2330:
2321:
2315:
2306:
2271:
2262:
2251:
2242:
2230:. Retrieved
2215:
2208:
2158:
2149:
2014:
1962:Zorn's lemma
1928:
1730:
1550:the numbers
1417:the numbers
1357:
1334:
1261:
1190:
1143:
989:real numbers
986:
782:
707:
655:
615:
601:
600:
596:Free algebra
594:
593:
587:
586:
555:
498:
460:
423:
392:
391:
371:Finite field
320:
267:Finite field
253:
252:
179:Initial ring
139:
138:
112:
111:
54:
29:
2161:represented
2006:cardinality
576:Simple ring
287:Jordan ring
161:Graded ring
43:Ring theory
2424:Categories
2298:1079.11002
2248:Green, Ben
2200:References
1411:Nesterenko
758:polynomial
582:Commutator
341:GCD domain
2411:MathWorld
2290:0938-0396
2232:April 11,
1873:α
1861:…
1847:α
1807:that are
1785:α
1778:…
1766:α
1679:π
1592:Γ
1563:π
1509:Γ
1472:Γ
1450:π
1425:π
1391:π
1343:π
1309:π
1278:π
1238:−
1168:π
1157:π
1098:π
1067:π
1043:are each
1025:π
1000:π
839:α
794:α
523:∞
301:Semifield
1929:Given a
1501:, where
987:The two
712:subfield
295:Semiring
281:Lie ring
63:Subrings
2359:0369110
2077:matroid
1833:, then
1521:is the
983:Example
755:trivial
710:over a
497:PrĂĽfer
99:•
2382:
2357:
2296:
2288:
2278:
2223:
2165:matrix
1464:, and
662:subset
149:Module
122:Kernel
2163:by a
1984:over
1811:over
939:over
895:over
855:over
685:field
683:of a
501:-ring
365:Field
261:Field
69:Ideal
56:Rings
2380:ISBN
2286:ISSN
2276:ISBN
2234:2008
2221:ISBN
1803:are
1731:The
1584:and
1355:and
1292:and
1084:and
1014:and
660:, a
2347:doi
2322:322
2294:Zbl
1191:not
1189:is
851:is
706:is
656:In
2426::
2408:.
2355:MR
2353:,
2341:,
2320:.
2292:.
2284:.
2156:.
1915:.
1437:,
1327:.
979:.
780:.
609:•
580:•
574:•
568:•
562:•
495:•
458:•
421:•
415:•
406:•
400:•
383:•
377:•
369:•
363:•
357:•
351:•
345:•
339:•
333:•
327:•
299:•
293:•
285:•
279:•
273:•
265:•
259:•
204:•
177:•
171:•
165:•
159:•
153:•
147:•
132:•
126:•
120:•
105:•
93:•
85:•
79:•
73:•
67:•
61:•
2414:.
2389:.
2362:.
2349::
2343:7
2300:.
2236:.
2175:K
2136:]
2133:T
2130:[
2127:K
2107:L
2087:T
2063:S
2043:L
2023:S
1992:K
1972:L
1948:K
1944:/
1940:L
1902:Q
1877:n
1868:e
1864:,
1858:,
1851:1
1842:e
1820:Q
1789:n
1781:,
1775:,
1770:1
1744:Q
1722:.
1709:Q
1684:n
1675:e
1654:n
1643:.
1630:Q
1609:)
1606:3
1602:/
1598:1
1595:(
1568:3
1559:e
1547:.
1534:Q
1489:)
1486:4
1482:/
1478:1
1475:(
1446:e
1397:e
1394:+
1370:Q
1358:e
1315:1
1312:+
1306:2
1303:=
1300:y
1273:=
1270:x
1247:1
1244:+
1241:y
1233:2
1229:x
1225:2
1222:=
1219:)
1216:y
1213:,
1210:x
1207:(
1204:P
1177:}
1174:1
1171:+
1165:2
1162:,
1152:{
1128:Q
1107:}
1104:1
1101:+
1095:2
1092:{
1072:}
1062:{
1031:1
1028:+
1022:2
967:S
947:K
923:K
903:K
883:S
863:K
817:K
797:}
791:{
768:K
741:S
721:K
694:L
671:S
645:e
638:t
631:v
528:)
519:p
515:(
511:Z
499:p
479:p
474:Q
461:p
442:p
437:Z
424:p
410:n
235:Z
231:1
227:/
222:Z
218:=
215:0
189:Z
20:)
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