2304:
1944:
2299:{\displaystyle {\begin{aligned}e^{2i\theta }&={\frac {e^{i\theta }}{e^{-i\theta }}}={\frac {\cos \theta +i\sin \theta }{\cos \theta -i\sin \theta }}={\frac {1+i\tan \theta }{1-i\tan \theta }}\\\theta &={\frac {1}{2i}}\ln \left({\frac {1+i\tan \theta }{1-i\tan \theta }}\right)\\\tan ^{-1}x&={\frac {1}{2i}}\ln \left({\frac {1+ix}{1-ix}}\right)\end{aligned}}}
1424:
A proof of
Liouville's theorem can be found in section 12.4 of Geddes, et al. See Lützen's scientific bibliography for a sketch of Liouville's original proof (Chapter IX. Integration in Finite Terms), its modern exposition and algebraic treatment (ibid. §61).
1397:
2325:
of a simple antiderivative is either trivial (if no field extension is required to express it), or is simply the additive group of the constants (corresponding to the constant of integration). Thus, an antiderivative's
1562:
743:
556:
985:
1217:
1421:) are those with this form. Thus, on an intuitive level, the theorem states that the only elementary antiderivatives are the "simple" functions plus a finite number of logarithms of "simple" functions.
305:
1939:
1949:
1275:
1881:
do not seem to satisfy the requirements of the theorem, since they are not (apparently) sums of rational functions and logarithms of rational functions. However, a calculation with
1751:
1797:
137:
1602:
1468:
1831:
1670:
1636:
1879:
1280:
1508:
2609:
88:
1702:
1084:
1058:
474:
167:
1113:
1401:
In other words, the only functions that have "elementary antiderivatives" (that is, antiderivatives living in, at worst, an elementary differential extension of
1032:
788:
676:
629:
348:
207:
2515:
1419:
1153:
1133:
1005:
932:
912:
880:
860:
836:
808:
765:
653:
606:
582:
497:
439:
415:
395:
367:
325:
237:
2666:
689:
502:
2394:
2330:
does not encode enough information to determine if it can be expressed using elementary functions, the major condition of
Liouville's theorem.
242:
2709:
2640:
2487:
1513:
937:
2558:
1158:
2766:
2370:
1887:
2761:
2632:
2776:
2771:
2507:
1941:
shows that in fact the antiderivatives can be written in the required manner (as logarithms of rational functions).
2358:
2314:
2327:
1222:
2474:. Studies in the History of Mathematics and Physical Sciences. Vol. 15. New York, NY: Springer New York.
2400:
1710:
1758:
2604:
2382:
2346:
839:
655:
is not necessarily equipped with a unique logarithm; one might adjoin many "logarithm-like" extensions to
561:
477:
101:
47:
1569:
1436:
211:
2588:
2572:
1802:
1641:
1607:
2376:
2352:
1836:
2364:
1488:
43:
36:
2340:
184:
172:
53:
1882:
2734:
2705:
2675:
2636:
2554:
2534:
2483:
1471:
747:
With the above caveat in mind, this element may be thought of as an exponential of an element
1675:
2600:
2584:
2568:
2524:
2475:
1063:
1037:
444:
142:
28:
2719:
2687:
2650:
171:
Liouville's theorem states that elementary antiderivatives, if they exist, are in the same
2715:
2701:
2683:
2646:
2388:
1089:
883:
418:
175:
as the function, plus possibly a finite number of applications of the logarithm function.
16:
Says when antiderivatives of elementary functions can be expressed as elementary functions
2737:
1014:
770:
658:
611:
330:
189:
2547:
1483:
1404:
1392:{\displaystyle f=c_{1}{\frac {Df_{1}}{f_{1}}}+\dotsb +c_{n}{\frac {Df_{n}}{f_{n}}}+Ds.}
1138:
1118:
990:
917:
897:
865:
845:
821:
793:
750:
638:
591:
567:
482:
424:
400:
380:
352:
310:
222:
91:
32:
2755:
2469:
2318:
2700:, Grundlehren der Mathematischen Wissenschaften , vol. 328, Berlin, New York:
2658:
2573:"Premier mémoire sur la détermination des intégrales dont la valeur est algébrique"
2322:
2589:"Second mémoire sur la détermination des intégrales dont la valeur est algébrique"
2529:
2626:
20:
2317:, but this is not strictly true. The theorem can be proved without any use of
2695:
2622:
2479:
1475:
632:
95:
2679:
2538:
2742:
585:
2385: – Integrals not expressible in closed-form from elementary functions
46:
cannot themselves be expressed as elementary functions. These are called
2605:"Note sur la détermination des intégrales dont la valeur est algébrique"
2403: – Analytic function that does not satisfy a polynomial equation
1557:{\displaystyle \operatorname {Con} (\mathbb {C} (x))=\mathbb {C} ,}
2379: – Elementary functions and their finitely iterated integrals
738:{\displaystyle {\frac {Dt}{t}}=Ds\quad {\text{ for some }}s\in F.}
551:{\displaystyle Dt={\frac {Ds}{s}}\quad {\text{ for some }}s\in F.}
2349: – Mathematical formula involving a given set of operations
980:{\displaystyle \operatorname {Con} (F)=\operatorname {Con} (G),}
886:
in the chain is either algebraic, logarithmic, or exponential.
2361: – Study of Galois symmetry groups of differential fields
1212:{\displaystyle c_{1},\ldots ,c_{n}\in \operatorname {Con} (F)}
90:
whose antiderivative is (with a multiplier of a constant) the
2545:
Geddes, Keith O.; Czapor, Stephen R.; Labahn, George (1992).
2631:, University Lecture Series, vol. 7, Providence, R.I.:
1474:
in a single variable has a derivation given by the standard
2313:
Liouville's theorem is sometimes presented as a theorem in
631:
in which case, this condition is analogous to the ordinary
300:{\displaystyle \operatorname {Con} (F)=\{f\in F:Df=0\}.}
1934:{\displaystyle e^{i\theta }=\cos \theta +i\sin \theta }
1763:
1580:
1479:
1008:
1947:
1890:
1839:
1805:
1761:
1713:
1678:
1644:
1610:
1572:
1516:
1491:
1439:
1407:
1283:
1225:
1161:
1141:
1121:
1092:
1066:
1040:
1017:
993:
940:
920:
900:
868:
848:
824:
796:
773:
753:
692:
661:
641:
614:
594:
570:
505:
485:
447:
427:
403:
383:
355:
333:
313:
245:
225:
192:
145:
104:
56:
2508:"Review of "Lectures on differential Galois theory""
686:
is a simple transcendental extension that satisfies
31:
in 1833 to 1841, places an important restriction on
2546:
2456:
2391: – Method for evaluating indefinite integrals
2298:
1933:
1873:
1825:
1791:
1745:
1696:
1664:
1630:
1596:
1556:
1502:
1462:
1413:
1391:
1269:
1211:
1147:
1127:
1107:
1078:
1052:
1026:
999:
979:
926:
906:
874:
854:
830:
802:
782:
759:
737:
670:
647:
623:
600:
576:
550:
491:
468:
433:
409:
389:
361:
342:
319:
299:
231:
201:
161:
131:
82:
2355: – Algebraic study of differential equations
2694:van der Put, Marius; Singer, Michael F. (2003),
27:, originally formulated by French mathematician
2610:Journal fĂĽr die reine und angewandte Mathematik
2697:Galois theory of linear differential equations
2516:Bulletin of the American Mathematical Society
8:
2667:Notices of the American Mathematical Society
2373: – System of arithmetic in proof theory
2309:Relationship with differential Galois theory
291:
264:
50:. A standard example of such a function is
2444:
2432:
2420:
1705:
1270:{\displaystyle f_{1},\ldots ,f_{n},s\in F}
2528:
2256:
2231:
2209:
2153:
2128:
2070:
2011:
1994:
1981:
1975:
1956:
1948:
1946:
1895:
1889:
1844:
1838:
1807:
1806:
1804:
1773:
1762:
1760:
1715:
1714:
1712:
1677:
1646:
1645:
1643:
1612:
1611:
1609:
1579:
1571:
1547:
1546:
1527:
1526:
1515:
1493:
1492:
1490:
1447:
1446:
1438:
1406:
1369:
1358:
1348:
1342:
1321:
1310:
1300:
1294:
1282:
1249:
1230:
1224:
1185:
1166:
1160:
1140:
1120:
1091:
1065:
1039:
1016:
992:
939:
919:
899:
867:
847:
823:
795:
772:
752:
718:
693:
691:
660:
640:
613:
593:
569:
531:
515:
504:
484:
446:
426:
402:
382:
354:
332:
312:
244:
224:
191:
150:
144:
105:
103:
69:
61:
55:
98:. Other examples include the functions
2413:
2628:Lectures on differential Galois theory
1746:{\displaystyle \mathbb {C} (x,\ln x).}
1792:{\displaystyle {\tfrac {1}{x^{2}+1}}}
7:
2395:Tarski's high school algebra problem
1478:with respect to that variable. The
132:{\displaystyle {\frac {\sin(x)}{x}}}
1799:does not have an antiderivative in
1638:does not have an antiderivative in
1597:{\displaystyle f:={\tfrac {1}{x}},}
1463:{\displaystyle F:=\mathbb {C} (x)}
14:
1009:elementary differential extension
814:elementary differential extension
2593:Journal de l'École Polytechnique
2577:Journal de l'École Polytechnique
2457:Geddes, Czapor & Labahn 1992
1826:{\displaystyle \mathbb {C} (x).}
1665:{\displaystyle \mathbb {C} (x).}
1631:{\displaystyle \mathbb {C} (x),}
564:. Intuitively, one may think of
2549:Algorithms for Computer Algebra
1874:{\displaystyle \tan ^{-1}(x)+C}
717:
530:
419:simple transcendental extension
42:The antiderivatives of certain
2553:. Kluwer Academic Publishers.
2371:Elementary function arithmetic
1862:
1856:
1817:
1811:
1737:
1719:
1656:
1650:
1622:
1616:
1540:
1537:
1531:
1523:
1457:
1451:
1206:
1200:
1135:contains an antiderivative of
971:
965:
953:
947:
838:if there is a finite chain of
463:
457:
307:Given two differential fields
258:
252:
120:
114:
1:
2633:American Mathematical Society
2530:10.1090/s0273-0979-96-00652-0
2367: – Mathematical function
2343: – Mathematical function
1503:{\displaystyle \mathbb {C} ;}
934:are differential fields with
48:nonelementary antiderivatives
2659:"Differential Galois theory"
2397: – Mathematical problem
83:{\displaystyle e^{-x^{2}},}
2793:
2471:Joseph Liouville 1809–1882
2359:Differential Galois theory
2315:differential Galois theory
1704:do, however, exist in the
2480:10.1007/978-1-4612-0989-8
2328:differential Galois group
1433:As an example, the field
35:that can be expressed as
1115:(in words, suppose that
2738:"Liouville's Principle"
2657:Magid, Andy R. (1999),
2468:LĂĽtzen, Jesper (1990).
2401:Transcendental function
1755:Likewise, the function
1697:{\displaystyle \ln x+C}
1480:constants of this field
560:This has the form of a
2767:Differential equations
2383:Nonelementary integral
2347:Closed-form expression
2300:
1935:
1875:
1827:
1793:
1747:
1698:
1666:
1632:
1598:
1558:
1504:
1464:
1415:
1393:
1271:
1213:
1149:
1129:
1109:
1080:
1079:{\displaystyle g\in G}
1054:
1053:{\displaystyle f\in F}
1028:
1001:
981:
928:
908:
876:
856:
832:
804:
784:
761:
739:
672:
649:
625:
602:
578:
562:logarithmic derivative
552:
493:
470:
469:{\displaystyle G=F(t)}
435:
411:
391:
363:
344:
321:
301:
233:
203:
163:
162:{\displaystyle x^{x}.}
133:
84:
2506:Bertrand, D. (1996),
2301:
1936:
1876:
1828:
1794:
1748:
1706:logarithmic extension
1699:
1667:
1633:
1599:
1559:
1505:
1465:
1416:
1394:
1272:
1214:
1155:). Then there exist
1150:
1130:
1110:
1081:
1055:
1029:
1002:
982:
929:
909:
877:
857:
833:
805:
785:
762:
740:
682:exponential extension
673:
650:
626:
603:
579:
553:
494:
471:
436:
412:
392:
373:logarithmic extension
364:
345:
322:
302:
234:
204:
164:
134:
85:
2762:Differential algebra
2595:. tome XIV: 149–193.
2579:. tome XIV: 124–148.
2377:Liouvillian function
2353:Differential algebra
2321:. Furthermore, the
1945:
1888:
1837:
1833:Its antiderivatives
1803:
1759:
1711:
1676:
1672:Its antiderivatives
1642:
1608:
1570:
1514:
1489:
1437:
1405:
1281:
1223:
1159:
1139:
1119:
1108:{\displaystyle Dg=f}
1090:
1064:
1038:
1015:
991:
938:
918:
898:
866:
846:
822:
794:
771:
751:
720: for some
690:
659:
639:
612:
592:
568:
533: for some
503:
483:
445:
425:
401:
381:
353:
331:
311:
243:
223:
190:
143:
102:
54:
44:elementary functions
37:elementary functions
2777:Theorems in algebra
2772:Field (mathematics)
2365:Elementary function
25:Liouville's theorem
2735:Weisstein, Eric W.
2341:Algebraic function
2296:
2294:
1931:
1871:
1823:
1789:
1787:
1743:
1694:
1662:
1628:
1594:
1589:
1554:
1500:
1472:rational functions
1460:
1411:
1389:
1267:
1209:
1145:
1125:
1105:
1076:
1050:
1027:{\displaystyle F.}
1024:
997:
977:
924:
904:
872:
852:
828:
800:
783:{\displaystyle F.}
780:
757:
735:
671:{\displaystyle F.}
668:
645:
624:{\displaystyle F,}
621:
598:
574:
548:
489:
466:
431:
407:
387:
359:
343:{\displaystyle G,}
340:
317:
297:
229:
202:{\displaystyle F,}
199:
185:differential field
173:differential field
159:
129:
80:
2711:978-3-540-44228-8
2642:978-0-8218-7004-4
2601:Liouville, Joseph
2585:Liouville, Joseph
2569:Liouville, Joseph
2489:978-1-4612-6973-1
2286:
2244:
2195:
2141:
2112:
2065:
2006:
1786:
1588:
1414:{\displaystyle F}
1375:
1327:
1148:{\displaystyle f}
1128:{\displaystyle G}
1000:{\displaystyle G}
927:{\displaystyle G}
907:{\displaystyle F}
875:{\displaystyle G}
855:{\displaystyle F}
831:{\displaystyle F}
803:{\displaystyle G}
760:{\displaystyle s}
721:
706:
648:{\displaystyle F}
601:{\displaystyle s}
577:{\displaystyle t}
534:
528:
492:{\displaystyle t}
434:{\displaystyle F}
410:{\displaystyle G}
390:{\displaystyle F}
362:{\displaystyle G}
320:{\displaystyle F}
232:{\displaystyle F}
127:
2784:
2748:
2747:
2722:
2690:
2674:(9): 1041–1049,
2663:
2653:
2618:
2596:
2580:
2564:
2552:
2541:
2532:
2512:
2494:
2493:
2465:
2459:
2454:
2448:
2442:
2436:
2430:
2424:
2418:
2305:
2303:
2302:
2297:
2295:
2291:
2287:
2285:
2271:
2257:
2245:
2243:
2232:
2217:
2216:
2200:
2196:
2194:
2174:
2154:
2142:
2140:
2129:
2113:
2111:
2091:
2071:
2066:
2064:
2038:
2012:
2007:
2005:
2004:
1989:
1988:
1976:
1967:
1966:
1940:
1938:
1937:
1932:
1903:
1902:
1880:
1878:
1877:
1872:
1852:
1851:
1832:
1830:
1829:
1824:
1810:
1798:
1796:
1795:
1790:
1788:
1785:
1778:
1777:
1764:
1752:
1750:
1749:
1744:
1718:
1703:
1701:
1700:
1695:
1671:
1669:
1668:
1663:
1649:
1637:
1635:
1634:
1629:
1615:
1604:which exists in
1603:
1601:
1600:
1595:
1590:
1581:
1563:
1561:
1560:
1555:
1550:
1530:
1509:
1507:
1506:
1501:
1496:
1469:
1467:
1466:
1461:
1450:
1420:
1418:
1417:
1412:
1398:
1396:
1395:
1390:
1376:
1374:
1373:
1364:
1363:
1362:
1349:
1347:
1346:
1328:
1326:
1325:
1316:
1315:
1314:
1301:
1299:
1298:
1276:
1274:
1273:
1268:
1254:
1253:
1235:
1234:
1218:
1216:
1215:
1210:
1190:
1189:
1171:
1170:
1154:
1152:
1151:
1146:
1134:
1132:
1131:
1126:
1114:
1112:
1111:
1106:
1085:
1083:
1082:
1077:
1059:
1057:
1056:
1051:
1033:
1031:
1030:
1025:
1006:
1004:
1003:
998:
986:
984:
983:
978:
933:
931:
930:
925:
913:
911:
910:
905:
881:
879:
878:
873:
861:
859:
858:
853:
837:
835:
834:
829:
816:
815:
809:
807:
806:
801:
789:
787:
786:
781:
766:
764:
763:
758:
744:
742:
741:
736:
722:
719:
707:
702:
694:
684:
683:
677:
675:
674:
669:
654:
652:
651:
646:
630:
628:
627:
622:
607:
605:
604:
599:
588:of some element
583:
581:
580:
575:
557:
555:
554:
549:
535:
532:
529:
524:
516:
498:
496:
495:
490:
475:
473:
472:
467:
440:
438:
437:
432:
416:
414:
413:
408:
396:
394:
393:
388:
375:
374:
368:
366:
365:
360:
349:
347:
346:
341:
326:
324:
323:
318:
306:
304:
303:
298:
239:is the subfield
238:
236:
235:
230:
216:
215:
208:
206:
205:
200:
168:
166:
165:
160:
155:
154:
138:
136:
135:
130:
128:
123:
106:
94:, familiar from
89:
87:
86:
81:
76:
75:
74:
73:
29:Joseph Liouville
2792:
2791:
2787:
2786:
2785:
2783:
2782:
2781:
2752:
2751:
2733:
2732:
2729:
2712:
2702:Springer-Verlag
2693:
2661:
2656:
2643:
2621:
2599:
2583:
2567:
2561:
2544:
2510:
2505:
2502:
2497:
2490:
2467:
2466:
2462:
2455:
2451:
2445:Liouville 1833c
2443:
2439:
2433:Liouville 1833b
2431:
2427:
2421:Liouville 1833a
2419:
2415:
2411:
2406:
2389:Risch algorithm
2336:
2311:
2293:
2292:
2272:
2258:
2252:
2236:
2224:
2205:
2202:
2201:
2175:
2155:
2149:
2133:
2121:
2115:
2114:
2092:
2072:
2039:
2013:
1990:
1977:
1968:
1952:
1943:
1942:
1891:
1886:
1885:
1883:Euler's formula
1840:
1835:
1834:
1801:
1800:
1769:
1768:
1757:
1756:
1709:
1708:
1674:
1673:
1640:
1639:
1606:
1605:
1568:
1567:
1512:
1511:
1487:
1486:
1484:complex numbers
1435:
1434:
1431:
1403:
1402:
1365:
1354:
1350:
1338:
1317:
1306:
1302:
1290:
1279:
1278:
1245:
1226:
1221:
1220:
1181:
1162:
1157:
1156:
1137:
1136:
1117:
1116:
1088:
1087:
1062:
1061:
1036:
1035:
1013:
1012:
989:
988:
936:
935:
916:
915:
896:
895:
892:
864:
863:
844:
843:
820:
819:
813:
812:
792:
791:
769:
768:
749:
748:
695:
688:
687:
681:
680:
657:
656:
637:
636:
610:
609:
590:
589:
566:
565:
517:
501:
500:
481:
480:
443:
442:
423:
422:
399:
398:
379:
378:
372:
371:
351:
350:
329:
328:
309:
308:
241:
240:
221:
220:
213:
212:
188:
187:
181:
146:
141:
140:
107:
100:
99:
65:
57:
52:
51:
33:antiderivatives
17:
12:
11:
5:
2790:
2788:
2780:
2779:
2774:
2769:
2764:
2754:
2753:
2750:
2749:
2728:
2727:External links
2725:
2724:
2723:
2710:
2691:
2654:
2641:
2623:Magid, Andy R.
2619:
2597:
2581:
2565:
2559:
2542:
2501:
2498:
2496:
2495:
2488:
2460:
2449:
2437:
2425:
2412:
2410:
2407:
2405:
2404:
2398:
2392:
2386:
2380:
2374:
2368:
2362:
2356:
2350:
2344:
2337:
2335:
2332:
2310:
2307:
2290:
2284:
2281:
2278:
2275:
2270:
2267:
2264:
2261:
2255:
2251:
2248:
2242:
2239:
2235:
2230:
2227:
2225:
2223:
2220:
2215:
2212:
2208:
2204:
2203:
2199:
2193:
2190:
2187:
2184:
2181:
2178:
2173:
2170:
2167:
2164:
2161:
2158:
2152:
2148:
2145:
2139:
2136:
2132:
2127:
2124:
2122:
2120:
2117:
2116:
2110:
2107:
2104:
2101:
2098:
2095:
2090:
2087:
2084:
2081:
2078:
2075:
2069:
2063:
2060:
2057:
2054:
2051:
2048:
2045:
2042:
2037:
2034:
2031:
2028:
2025:
2022:
2019:
2016:
2010:
2003:
2000:
1997:
1993:
1987:
1984:
1980:
1974:
1971:
1969:
1965:
1962:
1959:
1955:
1951:
1950:
1930:
1927:
1924:
1921:
1918:
1915:
1912:
1909:
1906:
1901:
1898:
1894:
1870:
1867:
1864:
1861:
1858:
1855:
1850:
1847:
1843:
1822:
1819:
1816:
1813:
1809:
1784:
1781:
1776:
1772:
1767:
1742:
1739:
1736:
1733:
1730:
1727:
1724:
1721:
1717:
1693:
1690:
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1627:
1624:
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1618:
1614:
1593:
1587:
1584:
1578:
1575:
1553:
1549:
1545:
1542:
1539:
1536:
1533:
1529:
1525:
1522:
1519:
1499:
1495:
1459:
1456:
1453:
1449:
1445:
1442:
1430:
1427:
1410:
1388:
1385:
1382:
1379:
1372:
1368:
1361:
1357:
1353:
1345:
1341:
1337:
1334:
1331:
1324:
1320:
1313:
1309:
1305:
1297:
1293:
1289:
1286:
1266:
1263:
1260:
1257:
1252:
1248:
1244:
1241:
1238:
1233:
1229:
1208:
1205:
1202:
1199:
1196:
1193:
1188:
1184:
1180:
1177:
1174:
1169:
1165:
1144:
1124:
1104:
1101:
1098:
1095:
1075:
1072:
1069:
1049:
1046:
1043:
1023:
1020:
996:
976:
973:
970:
967:
964:
961:
958:
955:
952:
949:
946:
943:
923:
903:
891:
888:
871:
851:
827:
799:
779:
776:
756:
734:
731:
728:
725:
716:
713:
710:
705:
701:
698:
678:Similarly, an
667:
664:
644:
620:
617:
597:
573:
547:
544:
541:
538:
527:
523:
520:
514:
511:
508:
488:
478:transcendental
465:
462:
459:
456:
453:
450:
430:
406:
386:
358:
339:
336:
316:
296:
293:
290:
287:
284:
281:
278:
275:
272:
269:
266:
263:
260:
257:
254:
251:
248:
228:
198:
195:
180:
177:
158:
153:
149:
126:
122:
119:
116:
113:
110:
92:error function
79:
72:
68:
64:
60:
15:
13:
10:
9:
6:
4:
3:
2:
2789:
2778:
2775:
2773:
2770:
2768:
2765:
2763:
2760:
2759:
2757:
2745:
2744:
2739:
2736:
2731:
2730:
2726:
2721:
2717:
2713:
2707:
2703:
2699:
2698:
2692:
2689:
2685:
2681:
2677:
2673:
2669:
2668:
2660:
2655:
2652:
2648:
2644:
2638:
2634:
2630:
2629:
2624:
2620:
2616:
2612:
2611:
2606:
2602:
2598:
2594:
2590:
2586:
2582:
2578:
2574:
2570:
2566:
2562:
2560:0-7923-9259-0
2556:
2551:
2550:
2543:
2540:
2536:
2531:
2526:
2522:
2518:
2517:
2509:
2504:
2503:
2499:
2491:
2485:
2481:
2477:
2473:
2472:
2464:
2461:
2458:
2453:
2450:
2446:
2441:
2438:
2434:
2429:
2426:
2422:
2417:
2414:
2408:
2402:
2399:
2396:
2393:
2390:
2387:
2384:
2381:
2378:
2375:
2372:
2369:
2366:
2363:
2360:
2357:
2354:
2351:
2348:
2345:
2342:
2339:
2338:
2333:
2331:
2329:
2324:
2320:
2319:Galois theory
2316:
2308:
2306:
2288:
2282:
2279:
2276:
2273:
2268:
2265:
2262:
2259:
2253:
2249:
2246:
2240:
2237:
2233:
2228:
2226:
2221:
2218:
2213:
2210:
2206:
2197:
2191:
2188:
2185:
2182:
2179:
2176:
2171:
2168:
2165:
2162:
2159:
2156:
2150:
2146:
2143:
2137:
2134:
2130:
2125:
2123:
2118:
2108:
2105:
2102:
2099:
2096:
2093:
2088:
2085:
2082:
2079:
2076:
2073:
2067:
2061:
2058:
2055:
2052:
2049:
2046:
2043:
2040:
2035:
2032:
2029:
2026:
2023:
2020:
2017:
2014:
2008:
2001:
1998:
1995:
1991:
1985:
1982:
1978:
1972:
1970:
1963:
1960:
1957:
1953:
1928:
1925:
1922:
1919:
1916:
1913:
1910:
1907:
1904:
1899:
1896:
1892:
1884:
1868:
1865:
1859:
1853:
1848:
1845:
1841:
1820:
1814:
1782:
1779:
1774:
1770:
1765:
1753:
1740:
1734:
1731:
1728:
1725:
1722:
1707:
1691:
1688:
1685:
1682:
1679:
1659:
1653:
1625:
1619:
1591:
1585:
1582:
1576:
1573:
1566:The function
1564:
1551:
1543:
1534:
1520:
1517:
1497:
1485:
1482:are just the
1481:
1477:
1473:
1454:
1443:
1440:
1428:
1426:
1422:
1408:
1399:
1386:
1383:
1380:
1377:
1370:
1366:
1359:
1355:
1351:
1343:
1339:
1335:
1332:
1329:
1322:
1318:
1311:
1307:
1303:
1295:
1291:
1287:
1284:
1264:
1261:
1258:
1255:
1250:
1246:
1242:
1239:
1236:
1231:
1227:
1203:
1197:
1194:
1191:
1186:
1182:
1178:
1175:
1172:
1167:
1163:
1142:
1122:
1102:
1099:
1096:
1093:
1073:
1070:
1067:
1047:
1044:
1041:
1021:
1018:
1010:
994:
974:
968:
962:
959:
956:
950:
944:
941:
921:
901:
890:Basic theorem
889:
887:
885:
869:
849:
841:
825:
817:
810:is called an
797:
777:
774:
754:
745:
732:
729:
726:
723:
714:
711:
708:
703:
699:
696:
685:
665:
662:
642:
634:
618:
615:
595:
587:
571:
563:
558:
545:
542:
539:
536:
525:
521:
518:
512:
509:
506:
486:
479:
460:
454:
451:
448:
428:
420:
404:
384:
376:
356:
337:
334:
314:
294:
288:
285:
282:
279:
276:
273:
270:
267:
261:
255:
249:
246:
226:
218:
217:
196:
193:
186:
178:
176:
174:
169:
156:
151:
147:
124:
117:
111:
108:
97:
93:
77:
70:
66:
62:
58:
49:
45:
40:
38:
34:
30:
26:
22:
2741:
2696:
2671:
2665:
2627:
2614:
2608:
2592:
2576:
2548:
2520:
2514:
2470:
2463:
2452:
2440:
2428:
2416:
2323:Galois group
2312:
1754:
1565:
1432:
1423:
1400:
893:
811:
746:
679:
635:. However,
559:
499:) such that
370:
369:is called a
210:
182:
170:
41:
24:
18:
882:where each
179:Definitions
21:mathematics
2756:Categories
2617:: 347–359.
2500:References
1476:derivative
1277:such that
633:chain rule
441:(that is,
96:statistics
2743:MathWorld
2680:0002-9920
2603:(1833c).
2587:(1833b).
2571:(1833a).
2539:0002-9904
2277:−
2250:
2219:
2211:−
2192:θ
2189:
2180:−
2172:θ
2169:
2147:
2119:θ
2109:θ
2106:
2097:−
2089:θ
2086:
2062:θ
2059:
2050:−
2047:θ
2044:
2036:θ
2033:
2021:θ
2018:
2002:θ
1996:−
1986:θ
1964:θ
1929:θ
1926:
1914:θ
1911:
1900:θ
1854:
1846:−
1732:
1683:
1521:
1510:that is,
1333:⋯
1262:∈
1240:…
1198:
1192:∈
1176:…
1071:∈
1045:∈
987:and that
963:
945:
884:extension
840:subfields
790:Finally,
727:∈
586:logarithm
540:∈
476:for some
271:∈
250:
214:constants
112:
63:−
2625:(1994),
2334:See also
1429:Examples
1086:satisfy
1034:Suppose
894:Suppose
183:For any
2720:1960772
2688:1710665
2651:1301076
584:as the
2718:
2708:
2686:
2678:
2649:
2639:
2557:
2537:
2486:
1007:is an
2662:(PDF)
2523:(2),
2511:(PDF)
2409:Notes
842:from
417:is a
2706:ISBN
2676:ISSN
2637:ISBN
2555:ISBN
2535:ISSN
2484:ISBN
1219:and
1060:and
914:and
327:and
209:the
139:and
2525:doi
2476:doi
2207:tan
2186:tan
2166:tan
2103:tan
2083:tan
2056:sin
2041:cos
2030:sin
2015:cos
1923:sin
1908:cos
1842:tan
1518:Con
1470:of
1195:Con
1011:of
960:Con
942:Con
862:to
818:of
767:of
608:of
421:of
397:if
377:of
247:Con
219:of
109:sin
19:In
2758::
2740:.
2716:MR
2714:,
2704:,
2684:MR
2682:,
2672:46
2670:,
2664:,
2647:MR
2645:,
2635:,
2615:10
2613:.
2607:.
2591:.
2575:.
2533:,
2521:33
2519:,
2513:,
2482:.
2247:ln
2144:ln
1729:ln
1680:ln
1577::=
1444::=
39:.
23:,
2746:.
2563:.
2527::
2492:.
2478::
2447:.
2435:.
2423:.
2289:)
2283:x
2280:i
2274:1
2269:x
2266:i
2263:+
2260:1
2254:(
2241:i
2238:2
2234:1
2229:=
2222:x
2214:1
2198:)
2183:i
2177:1
2163:i
2160:+
2157:1
2151:(
2138:i
2135:2
2131:1
2126:=
2100:i
2094:1
2080:i
2077:+
2074:1
2068:=
2053:i
2027:i
2024:+
2009:=
1999:i
1992:e
1983:i
1979:e
1973:=
1961:i
1958:2
1954:e
1920:i
1917:+
1905:=
1897:i
1893:e
1869:C
1866:+
1863:)
1860:x
1857:(
1849:1
1821:.
1818:)
1815:x
1812:(
1808:C
1783:1
1780:+
1775:2
1771:x
1766:1
1741:.
1738:)
1735:x
1726:,
1723:x
1720:(
1716:C
1692:C
1689:+
1686:x
1660:.
1657:)
1654:x
1651:(
1647:C
1626:,
1623:)
1620:x
1617:(
1613:C
1592:,
1586:x
1583:1
1574:f
1552:,
1548:C
1544:=
1541:)
1538:)
1535:x
1532:(
1528:C
1524:(
1498:;
1494:C
1458:)
1455:x
1452:(
1448:C
1441:F
1409:F
1387:.
1384:s
1381:D
1378:+
1371:n
1367:f
1360:n
1356:f
1352:D
1344:n
1340:c
1336:+
1330:+
1323:1
1319:f
1312:1
1308:f
1304:D
1296:1
1292:c
1288:=
1285:f
1265:F
1259:s
1256:,
1251:n
1247:f
1243:,
1237:,
1232:1
1228:f
1207:)
1204:F
1201:(
1187:n
1183:c
1179:,
1173:,
1168:1
1164:c
1143:f
1123:G
1103:f
1100:=
1097:g
1094:D
1074:G
1068:g
1048:F
1042:f
1022:.
1019:F
995:G
975:,
972:)
969:G
966:(
957:=
954:)
951:F
948:(
922:G
902:F
870:G
850:F
826:F
798:G
778:.
775:F
755:s
733:.
730:F
724:s
715:s
712:D
709:=
704:t
700:t
697:D
666:.
663:F
643:F
619:,
616:F
596:s
572:t
546:.
543:F
537:s
526:s
522:s
519:D
513:=
510:t
507:D
487:t
464:)
461:t
458:(
455:F
452:=
449:G
429:F
405:G
385:F
357:G
338:,
335:G
315:F
295:.
292:}
289:0
286:=
283:f
280:D
277::
274:F
268:f
265:{
262:=
259:)
256:F
253:(
227:F
197:,
194:F
157:.
152:x
148:x
125:x
121:)
118:x
115:(
78:,
71:2
67:x
59:e
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