Galois theory - Knowledge

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possible to solve some equations, including all those of degree four or lower, in the above manner, and why it is not possible for most equations of degree five or higher. Furthermore, it provides a means of determining whether a particular equation can be solved that is both conceptually clear and

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is a symmetric function in the roots that reflects properties of the roots – it is zero if and only if the polynomial has a multiple root, and for quadratic and cubic polynomials it is positive if and only if all roots are real and distinct, and negative if and only if there is a pair of distinct

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Does there exist a formula for the roots of a fifth (or higher) degree polynomial equation in terms of the coefficients of the polynomial, using only the usual algebraic operations (addition, subtraction, multiplication, division) and application of radicals (square roots, cube roots,

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Galois theory implies that, since the polynomial is irreducible, the Galois group has at least four elements. For proving that the Galois group consists of these four permutations, it suffices thus to show that every element of the Galois group is determined by the image of

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accompanied by some of his own explanations. Prior to this publication, Liouville announced Galois' result to the Academy in a speech he gave on 4 July 1843. According to Allan Clark, Galois's characterization "dramatically supersedes the work of Abel and Ruffini."

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a memoir on his theory of solvability by radicals; Galois' paper was ultimately rejected in 1831 as being too sketchy and for giving a condition in terms of the roots of the equation instead of its coefficients. Galois then died in a duel in 1832, and his paper,

1481: 344:... the first person who understood the general doctrine of the formation of the coefficients of the powers from the sum of the roots and their products. He was the first who discovered the rules for summing the powers of the roots of any equation. 395:

at his disposal, nor the algebraic notation to be able to describe a general cubic equation. With the benefit of modern notation and complex numbers, the formulae in this book do work in the general case, but Cardano did not know this. It was

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Galois' theory was notoriously difficult for his contemporaries to understand, especially to the level where they could expand on it. For example, in his 1846 commentary, Liouville completely missed the group-theoretic core of Galois' method.

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of the roots, which yielded an auxiliary polynomial of lower degree, providing a unified understanding of the solutions and laying the groundwork for group theory and Galois' theory. Crucially, however, he did not consider

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It is known that a Galois group modulo a prime is isomorphic to a subgroup of the Galois group over the rationals. A permutation group on 5 objects with elements of orders 6 and 5 must be the symmetric group

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It removes the rather artificial reliance on chasing roots of polynomials. That is, different polynomials may yield the same extension fields, and the modern approach recognizes the connection between these

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failed to grasp its depth and popular British algebra textbooks did not even mention Galois' theory until well after the turn of the century. In Germany, Kronecker's writings focused more on Abel's result.

480:. This group was always solvable for polynomials of degree four or less, but not always so for polynomials of degree five and greater, which explains why there is no general solution in higher degrees. 180:

provides a counterexample proving that there are polynomial equations for which such a formula cannot exist. Galois' theory provides a much more complete answer to this question, by explaining why it

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construction. It gives an elegant characterization of the ratios of lengths that can be constructed with this method. Using this, it becomes relatively easy to answer such classical problems of

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allows one to determine whether a polynomial is solvable in radicals, depending on whether its Galois group has the property of solvability. In essence, each field extension

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The birth and development of Galois theory was caused by the following question, which was one of the main open mathematical questions until the beginning of 19th century:

3175: 2132: 2067: 438:, not just a single permutation. His solution contained a gap, which Cauchy considered minor, though this was not patched until the work of the Norwegian mathematician 1946:{\displaystyle {\begin{aligned}\varphi (B)&={\frac {-1}{\varphi (A)}},\\\varphi (C)&={\frac {1}{\varphi (A)}},\\\varphi (D)&=-\varphi (A).\end{aligned}}} 892: 364:, who did not however publish his results; this method, though, only solved one type of cubic equation. This solution was then rediscovered independently in 1535 by 927: 336:, the expression of coefficients of a polynomial in terms of the roots (not only for positive roots) was first understood by the 17th-century French mathematician 1648:

of these four roots, four are particularly simple, those consisting in the sign change of 0, 1, or 2 square roots. They form a group that is isomorphic to the

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but without the proof that the list of constructible polygons was complete; all known proofs that this characterization is complete require Galois theory).

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by adjoining the positive square roots of 2 and 3, together with its subfields; on the right, the corresponding lattice diagram of their Galois groups.

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It allows one to more easily study infinite extensions. Again this is important in algebraic number theory, where for example one often discusses the

472:, who showed that whether a polynomial was solvable or not was equivalent to whether or not the permutation group of its roots – in modern terms, its 1696: 2877:, the problem is not very difficult, and all finite groups do occur as Galois groups. For showing this, one may proceed as follows. Choose a field 2638:), and a systematic way for testing whether a specific polynomial is solvable by radicals. The Abel–Ruffini theorem results from the fact that for 3864: 376:. After the discovery of del Ferro's work, he felt that Tartaglia's method was no longer secret, and thus he published his solution in his 1545 4136: 4071: 4036: 3732: 3690: 3635: 3610: 3576: 2305:

The connection between the two approaches is as follows. The coefficients of the polynomial in question should be chosen from the base field

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Galois theory has been used to solve classic problems including showing that two problems of antiquity cannot be solved as they were stated (

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modulo 3 has no linear or quadratic factor, and hence is irreducible. Thus its modulo 3 Galois group contains an element of order 5.

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after the roots have been permuted. Originally, the theory had been developed for algebraic equations whose coefficients are

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of permutations. Lagrange's method did not extend to quintic equations or higher, because the resolvent had higher degree.

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In this book, however, Cardano did not provide a "general formula" for the solution of a cubic equation, as he had neither

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On the other hand, it is an open problem whether every finite group is the Galois group of a field extension of the field

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fourteen years after his death. The theory took longer to become popular among mathematicians and to be well understood.

378: 365: 3335: 72:, allows reducing certain problems in field theory to group theory, which makes them simpler and easier to understand. 3957: 3506: 3196: 2323:. Any permutation of the roots which respects algebraic equations as described above gives rise to an automorphism of 1104: 3093:) of the 26 sporadic simple groups. There is even a polynomial with integral coefficients whose Galois group is the 4128: 4028: 3465: 3442: 3482: 3477: 497: 3437:

showed that this establishes a one-to-one correspondence. The condition imposed by Jacobson has been removed by

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The Genesis of the Abstract Group Concept: A Contribution to the History of the Origin of Abstract Group Theory

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wrote little about Galois' theory, but lectured on it at Göttingen in 1858, showing a very good understanding.

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who attended some of Liouville's talks, included Galois' theory in his 1866 (third edition) of his textbook

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The members of the Galois group must preserve any algebraic equation with rational coefficients involving

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who managed to understand how to work with complex numbers in order to solve all forms of cubic equation.

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should be the field obtained by adjoining the roots of the polynomial in question to the base field

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in either of the last two equations we obtain another true statement. For example, the equation

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which are not solvable by radicals (this was proven independently, using a similar method, by

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are exchanged. However, this relation is not considered here, because it has the coefficient

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If all the factor groups in its composition series are cyclic, the Galois group is called

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This is one of the simplest examples of a non-solvable quintic polynomial. According to

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There are several advantages to the modern approach over the permutation group approach.

535:. Outside France, Galois' theory remained more obscure for a longer period. In Britain, 4121: 3970:

Jacobson, Nathan (1944), "Galois theory of purely inseparable fields of exponent one",

3923:(1930). "A short account of the history of symmetric functions of roots of equations". 3719:. Algorithms and Computation in Mathematics. Vol. 11. Springer. pp. 181–218. 2522: 528: 477: 373: 333: 17: 4183: 4212: 3834: 3094: 3082: 2574: 536: 337: 2689: 1461:{\displaystyle (x^{2}-1)^{2}-8x^{2}=(x^{2}-1-2x{\sqrt {2}})(x^{2}-1+2x{\sqrt {2}}).} 468:

quintic or higher polynomial could be determined to be solvable or not was given by

4044: 4018:(Chapter 4 gives an introduction to the field-theoretic approach to Galois theory.) 3896: 3454: 3122: 3078: 2874: 2660: 2548: 2540: 2526: 2447: 2443: 2299: 2283: 987: 685: 473: 354: 349: 65: 4162: 3652: 3811:. The Student Mathematical Library. Vol. 35. American Mathematical Society. 3077:. Various people have solved the inverse Galois problem for selected non-Abelian 3071:

proved that every solvable finite group is the Galois group of some extension of

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is algebraic, but not expressible in terms of radicals. The other four roots are

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Galois' theory also gives a clear insight into questions concerning problems in

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The cubic was first partly solved by the 15–16th-century Italian mathematician

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van der Waerden, Modern Algebra (1949 English edn.), Vol. 1, Section 61, p.191

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Given a polynomial, it may be that some of the roots are connected by various

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of the polynomial, which is explicitly described in the following examples.

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satisfying the Leibniz rule. In this correspondence, an intermediate field

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zero, but nonzero characteristic arises frequently in number theory and in

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The quintic was almost proven to have no general solutions by radicals by

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One of the great triumphs of Galois Theory was the proof that for every

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are the elementary polynomials of degree 0, 1 and 2 in two variables.

4171:(Later republished in English by Springer under the title "Algebra".) 3121:, there is a Galois theory where the Galois group is replaced by the 503:

Mémoire sur les conditions de résolubilité des équations par radicaux

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Richard Dedekind 1831–1981; eine Würdigung zu seinem 150. Geburtstag

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of the Galois group. If a factor group in the composition series is

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This was first formalized by the 16th-century French mathematician

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This implies that the permutation is well defined by the image of

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is a permutation that belongs to the Galois group, we must have:

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solved the quartic polynomial; his solution was also included in

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Scharlau, Winfried; Dedekind, Ilse; Dedekind, Richard (1981).

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yields another true relation. This results from the theory of

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Mathematical connection between field theory and group theory

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Purely Inseparable Galois theory I: The Fundamental Theorem

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Prasolov, V.V. (2004). "5 Galois Theory Theorem 5.4.5(a)".

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In the first example above, we were studying the extension

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if its roots may be expressed by a formula involving only

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In the form mentioned above, including in particular the

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is crucial in many areas of mathematics. For example, in

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A similar discussion applies to any quadratic polynomial

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is to find a field extension with a given Galois group.

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This implies that the Galois group is isomorphic to the

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satisfying appropriate further conditions is mapped to

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Galois Theory for Beginners: A Historical Perspective

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modulo 2 factors into polynomials of orders 2 and 3,

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Traité des substitutions et des équations algébriques

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Réflexions sur la résolution algébrique des équations

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We conclude that the Galois group of the polynomial

657:. The central idea of Galois' theory is to consider 2759:Neither does it have linear factors modulo 2 or 3. 1475:to each factor, one sees that the four roots are 496:In 1830 Galois (at the age of 18) submitted to the 4161: 4120: 3895: 3425: 3392: 3324: 3267: 3169: 2126: 2061: 1945: 1772: 1633: 1460: 1307: 1169:If the polynomial has rational roots, for example 921: 886: 833: 742: 3393:{\displaystyle \{x\in F,f(x)=0\ \forall f\in V\}} 3268:{\displaystyle Der_{E}(F,F)\subset Der_{K}(F,F)} 966:. It is more generally true that this holds for 3438: 169: 137:(this characterization was previously given by 4047:, and some generalisations, leading to Galois 3441:, by giving a correspondence using notions of 2557:, and if in the corresponding field extension 3685:. European Mathematical Society. p. 10. 3468:for a Galois theory of differential equations 1321:in an unusual way, it can also be written as 844:Examples of algebraic equations satisfied by 665:algebraic equation satisfied by the roots is 8: 3683:The Mathematical Writings of Évariste Galois 3681:Galois, Évariste; Neumann, Peter M. (2011). 3387: 3339: 449:While Ruffini and Abel established that the 2225:. Unsourced material may be challenged and 661:(or rearrangements) of the roots such that 598:. Unsourced material may be challenged and 492:A portrait of Évariste Galois aged about 15 75:Galois introduced the subject for studying 3533: 3474:for a vast generalization of Galois theory 2425:It permits a far simpler statement of the 2257:In the modern approach, one starts with a 994:; that is, in any such relation, swapping 250:Galois' theory originated in the study of 241:Abstract algebra § Early group theory 3851: 3501: 3499: 3411: 3405: 3337: 3301: 3283: 3244: 3210: 3198: 3146: 3134: 2899:is (up to isomorphism) a subgroup of the 2873:As long as one does not also specify the 2830:, which is therefore the Galois group of 2245:Learn how and when to remove this message 2117: 2115: 2052: 2050: 1874: 1822: 1799: 1797: 1700: 1698: 1617: 1607: 1580: 1570: 1543: 1533: 1509: 1499: 1485: 1483: 1445: 1424: 1407: 1386: 1370: 1354: 1338: 1329: 1293: 1277: 1271: 1123:, this Galois group is isomorphic to the 905: 864: 817: 787: 767: 765: 713: 707: 618:Learn how and when to remove this message 548:'s books of the 1880s, based on Jordan's 4123:Groups as Galois groups: an introduction 4023:Janelidze, G.; Borceux, Francis (2001). 3434: 2517:Solvable groups and solution by radicals 29: 3544: 3495: 2773:modulo 2 is cyclic of order 6, because 464:, and the precise criterion by which a 83:. This allowed him to characterize the 3842:Brantner, Lukas; Waldron, Joe (2020), 2302:for further explanation and examples. 1119:. As all groups with two elements are 1029:are related by the algebraic equation 431:in 1799, whose key insight was to use 151:Galois theory has been generalized to 3563:Galois' Theory of Algebraic Equations 3325:{\displaystyle V\subset Der_{K}(F,F)} 3113:. For a purely inseparable extension 2442:, one often does Galois theory using 2175:(change of sign of both square roots) 7: 3461:Fundamental theorem of Galois theory 3107:fundamental theorem of Galois theory 3040:fundamental theorem of Galois theory 3010:, by a basic result of Emil Artin. 2624:, there exist polynomials of degree 2467:, defined to be the Galois group of 2427:fundamental theorem of Galois theory 2223:adding citations to reliable sources 596:adding citations to reliable sources 407:by the French-Italian mathematician 70:fundamental theorem of Galois theory 680:These permutations together form a 38:diagram of the field obtained from 4168:. New York: Frederick Ungar. 1949. 3375: 3111:purely inseparable field extension 2432:The use of base fields other than 1087:consists of two permutations: the 1047:, which does not remain true when 453:quintic could not be solved, some 403:A further step was the 1770 paper 230:not possible with the same method? 25: 1660:, which can be shown as follows. 757:, we find that the two roots are 163:Application to classical problems 4196: 4182: 4147:van der Waerden, Bartel Leendert 2756:, this has no rational zeroes. 2589:can then be expressed using the 2195: 1690:Among these equations, we have: 1308:{\displaystyle x^{4}-10x^{2}+1.} 568: 457:quintics can be solved, such as 355:Discriminant:Nature of the roots 264:elementary symmetric polynomials 60:, provides a connection between 4005:(2nd ed.). W. H. Freeman. 2634:a few years before, and is the 2496:It allows for consideration of 2187:Modern approach by field theory 4153:(in German). Berlin: Springer. 3485:, a sub-field of Galois theory 3426:{\displaystyle F^{p}\subset K} 3363: 3357: 3319: 3307: 3262: 3250: 3228: 3216: 3164: 3152: 2685:A non-solvable quintic example 2282:"), and examines the group of 2173: 2108: 2043: 2002: 1933: 1927: 1911: 1905: 1889: 1883: 1864: 1858: 1842: 1836: 1812: 1806: 1452: 1417: 1414: 1379: 1351: 1331: 144:Galois' work was published by 114:. This widely generalizes the 95:of their roots—an equation is 91:in terms of properties of the 1: 4187:The dictionary definition of 3925:American Mathematical Monthly 3567:. World Scientific. pp. 3439:Brantner & Waldron (2020) 743:{\displaystyle x^{2}-4x+1=0.} 353:complex conjugate roots. See 4083:Foundations of Galois Theory 3603:Elements of Abstract Algebra 3472:Grothendieck's Galois theory 3170:{\displaystyle Der_{K}(F,F)} 3022:by restriction of action of 1107:permutation which exchanges 266:in the roots. For instance, 157:Grothendieck's Galois theory 3958:Encyclopedia of Mathematics 3725:10.1007/978-3-642-03980-5_5 2595:th root of some element of 2377:is the field obtained from 2127:{\displaystyle {\sqrt {2}}} 2062:{\displaystyle {\sqrt {3}}} 556:'s 1895 algebra textbook. 56:, originally introduced by 4235: 4159:(of 2nd revised edition): 4129:Cambridge University Press 4104:(2nd ed.). Springer. 4029:Cambridge University Press 3872:(in Latin). Archived from 3591:Stewart, 3rd ed., p. xxiii 3466:Differential Galois theory 3443:derived algebraic geometry 2859: 2852:was fond of this example. 560:Permutation group approach 525:Cours d'algèbre supérieure 238: 221:compass and a straightedge 215:Why is it not possible to 129:), and characterizing the 4119:Völklein, Helmut (1996). 4081:Postnikov, M. M. (2004). 3478:Topological Galois theory 3067:of the rational numbers. 1091:permutation which leaves 498:Paris Academy of Sciences 366:Niccolò Fontana Tartaglia 3660:. Braunschweig: Vieweg. 3630:. Courier. p. 118. 3605:. Courier. p. 131. 2914:. Choose indeterminates 1263:Consider the polynomial 254:– the coefficients of a 194:compass and straightedge 4100:Rotman, Joseph (1998). 4060:Algebraic Number Theory 3754:Algebraic Number Theory 3400:. Under the assumption 2994:. The Galois group of 2927:, one for each element 2440:algebraic number theory 1228:is no longer true when 483: 185:easily expressed as an 68:. This connection, the 18:Solvability by radicals 4085:. Dover Publications. 3626:Wussing, Hans (2007). 3601:Clark, Allan (1984) . 3427: 3394: 3326: 3269: 3171: 3101:Inseparable extensions 3042:, the Galois group of 2868:inverse Galois problem 2862:Inverse Galois problem 2856:Inverse Galois problem 2724: 2498:inseparable extensions 2128: 2063: 1947: 1774: 1644:Among the 24 possible 1635: 1462: 1309: 1165:are rational numbers. 1017:One might object that 923: 888: 887:{\displaystyle A+B=4,} 835: 744: 493: 346: 174: 45: 3483:Artin–Schreier theory 3428: 3395: 3327: 3270: 3172: 2939:, and adjoin them to 2754:rational root theorem 2730:cites the polynomial 2712:, the lone real root 2692: 2459:absolute Galois group 2298:. See the article on 2129: 2064: 1948: 1775: 1636: 1463: 1319:Completing the square 1310: 1008:symmetric polynomials 924: 922:{\displaystyle AB=1.} 889: 836: 745: 554:Heinrich Martin Weber 491: 409:Joseph Louis Lagrange 368:, who shared it with 342: 112:arithmetic operations 110:, and the four basic 33: 4205:at Wikimedia Commons 4062:. Berlin, New York: 3513:. Chapman and Hall. 3404: 3336: 3282: 3197: 3183:linear endomorphisms 3133: 2762:The Galois group of 2636:Abel–Ruffini theorem 2573:already contains a 2219:improve this section 2114: 2049: 1796: 1783:It follows that, if 1697: 1482: 1328: 1270: 1125:multiplicative group 904: 863: 764: 706: 592:improve this section 521:Joseph Alfred Serret 444:Abel–Ruffini theorem 178:Abel–Ruffini theorem 127:trisecting the angle 116:Abel–Ruffini theorem 97:solvable by radicals 89:solvable by radicals 85:polynomial equations 4157:English translation 3921:Funkhouser, H. Gray 3902:. Springer-Verlag. 3557:Tignol, Jean-Pierre 2965:. Contained within 2908:on the elements of 2883:and a finite group 2693:For the polynomial 2110:(change of sign of 2045:(change of sign of 1247:roots, for example 1103:untouched, and the 632:algebraic equations 413:Lagrange resolvents 411:, in his method of 252:symmetric functions 217:trisect every angle 3892:Edwards, Harold M. 3423: 3390: 3322: 3265: 3167: 3032:of this action is 2979:rational functions 2725: 2545:composition series 2506:algebraic geometry 2454:as the base field. 2333:, and vice versa. 2124: 2059: 1943: 1941: 1770: 1768: 1631: 1629: 1458: 1305: 972:algebraic relation 919: 884: 831: 829: 740: 698:quadratic equation 692:Quadratic equation 684:, also called the 527:. Serret's pupil, 494: 362:Scipione del Ferro 348:In this vein, the 153:Galois connections 46: 4201:Media related to 4138:978-0-521-56280-5 4073:978-0-387-94225-4 4038:978-0-521-80309-0 3861:Cardano, Gerolamo 3805:Bewersdorff, Jörg 3734:978-3-642-03979-9 3692:978-3-03719-104-0 3637:978-0-486-45868-7 3612:978-0-486-14035-3 3578:978-981-02-4541-2 3457:for more examples 3374: 2945:to get the field 2669:alternating group 2632:Niels Henrik Abel 2539:corresponds to a 2485:algebraic closure 2255: 2254: 2247: 2122: 2057: 1893: 1846: 1622: 1612: 1585: 1575: 1548: 1538: 1514: 1504: 1473:quadratic formula 1450: 1412: 822: 792: 755:quadratic formula 682:permutation group 628: 627: 620: 440:Niels Henrik Abel 340:; Hutton writes: 228:doubling the cube 123:doubling the cube 93:permutation group 34:On the left, the 16:(Redirected from 4226: 4200: 4186: 4169: 4167: 4154: 4142: 4126: 4115: 4096: 4077: 4042: 4016: 3999:Jacobson, Nathan 3994: 3966: 3948: 3913: 3901: 3887: 3885: 3884: 3878: 3871: 3856: 3855: 3838: 3817:10.1090/stml/035 3800: 3769: 3768: 3745: 3739: 3738: 3712: 3706: 3703: 3697: 3696: 3678: 3672: 3671: 3659: 3648: 3642: 3641: 3623: 3617: 3616: 3598: 3592: 3589: 3583: 3582: 3566: 3553: 3547: 3542: 3536: 3531: 3525: 3524: 3503: 3432: 3430: 3429: 3424: 3416: 3415: 3399: 3397: 3396: 3391: 3372: 3331: 3329: 3328: 3323: 3306: 3305: 3275:. Conversely, a 3274: 3272: 3271: 3266: 3249: 3248: 3215: 3214: 3176: 3174: 3173: 3168: 3151: 3150: 3092: 3076: 3069:Igor Shafarevich 3066: 3057: 3051: 3037: 3027: 3021: 3015: 3009: 3003: 2993: 2976: 2970: 2964: 2944: 2938: 2932: 2926: 2913: 2907: 2898: 2891:Cayley's theorem 2888: 2882: 2840: 2829: 2816: 2803: 2783: 2772: 2748: 2718: 2711: 2680: 2658: 2644: 2629: 2623: 2613: 2600: 2594: 2588: 2581:th root of unity 2580: 2572: 2566: 2556: 2538: 2521:The notion of a 2492: 2482: 2476: 2466: 2437: 2417: 2391: 2390: 2389: 2382: 2376: 2374: 2373: 2361:rational numbers 2359:is the field of 2358: 2352: 2347: 2346: 2332: 2322: 2316: 2311:. The top field 2310: 2297: 2291: 2281: 2275: 2269: 2250: 2243: 2239: 2236: 2230: 2199: 2191: 2181:Klein four-group 2174: 2171: 2133: 2131: 2130: 2125: 2123: 2118: 2109: 2106: 2068: 2066: 2065: 2060: 2058: 2053: 2044: 2041: 2003: 2000: 1961: 1952: 1950: 1949: 1944: 1942: 1894: 1892: 1875: 1847: 1845: 1831: 1823: 1788: 1779: 1777: 1776: 1771: 1769: 1686: 1680: 1674: 1668: 1659: 1650:Klein four-group 1640: 1638: 1637: 1632: 1630: 1623: 1618: 1613: 1608: 1586: 1581: 1576: 1571: 1549: 1544: 1539: 1534: 1515: 1510: 1505: 1500: 1471:By applying the 1467: 1465: 1464: 1459: 1451: 1446: 1429: 1428: 1413: 1408: 1391: 1390: 1375: 1374: 1359: 1358: 1343: 1342: 1314: 1312: 1311: 1306: 1298: 1297: 1282: 1281: 1259:Quartic equation 1253: 1239: 1233: 1227: 1216: 1209: 1202: 1183: 1164: 1158: 1152: 1146: 1129: 1118: 1112: 1102: 1096: 1086: 1068: 1067: 1066: 1058: 1052: 1046: 1044: 1043: 1028: 1022: 1012:binomial theorem 1005: 999: 985: 979: 965: 954: 943: 937: 928: 926: 925: 920: 893: 891: 890: 885: 855: 849: 840: 838: 837: 832: 830: 823: 818: 793: 788: 749: 747: 746: 741: 718: 717: 671:rational numbers 656: 645: 639: 623: 616: 612: 609: 603: 572: 564: 507:Joseph Liouville 484:Galois' writings 463: 385:Lodovico Ferrari 374:Cardano's method 370:Gerolamo Cardano 330:Viète's formulas 320: 314: 304: 256:monic polynomial 206:regular polygons 146:Joseph Liouville 131:regular polygons 107: 43: 21: 4234: 4233: 4229: 4228: 4227: 4225: 4224: 4223: 4209: 4208: 4179: 4160: 4151:Moderne Algebra 4145: 4139: 4118: 4112: 4099: 4093: 4080: 4074: 4064:Springer-Verlag 4054: 4039: 4025:Galois Theories 4022: 4013: 4003:Basic Algebra I 3997: 3984:10.2307/2371772 3969: 3953:"Galois theory" 3951: 3937:10.2307/2299273 3919: 3910: 3890: 3882: 3880: 3876: 3869: 3859: 3841: 3827: 3803: 3797: 3781: 3778: 3773: 3772: 3765: 3747: 3746: 3742: 3735: 3714: 3713: 3709: 3704: 3700: 3693: 3680: 3679: 3675: 3668: 3657: 3650: 3649: 3645: 3638: 3625: 3624: 3620: 3613: 3600: 3599: 3595: 3590: 3586: 3579: 3555: 3554: 3550: 3543: 3539: 3534:Funkhouser 1930 3532: 3528: 3521: 3505: 3504: 3497: 3492: 3451: 3435:Jacobson (1944) 3407: 3402: 3401: 3334: 3333: 3297: 3280: 3279: 3240: 3206: 3195: 3194: 3142: 3131: 3130: 3103: 3091: 3085: 3072: 3062: 3053: 3043: 3038:, then, by the 3033: 3023: 3017: 3011: 3005: 2995: 2991: 2982: 2972: 2966: 2962: 2946: 2940: 2934: 2928: 2924: 2915: 2909: 2903: 2901:symmetric group 2894: 2884: 2878: 2864: 2858: 2831: 2828: 2822: 2807: 2785: 2774: 2763: 2731: 2728:Van der Waerden 2721:complex numbers 2713: 2694: 2687: 2679: 2671: 2665:normal subgroup 2657: 2649: 2647:symmetric group 2639: 2625: 2618: 2609: 2596: 2590: 2584: 2576: 2568: 2558: 2552: 2530: 2519: 2488: 2478: 2468: 2462: 2433: 2393: 2387: 2385: 2384: 2378: 2371: 2369: 2364: 2354: 2344: 2342: 2337: 2324: 2318: 2312: 2306: 2293: 2287: 2277: 2271: 2261: 2259:field extension 2251: 2240: 2234: 2231: 2216: 2200: 2189: 2172: 2137: 2112: 2111: 2107: 2072: 2047: 2046: 2042: 2007: 2001: 1966: 1957: 1940: 1939: 1914: 1899: 1898: 1879: 1867: 1852: 1851: 1832: 1824: 1815: 1794: 1793: 1784: 1767: 1766: 1756: 1744: 1743: 1733: 1724: 1723: 1710: 1695: 1694: 1682: 1676: 1670: 1664: 1657: 1628: 1627: 1597: 1591: 1590: 1560: 1554: 1553: 1526: 1520: 1519: 1492: 1480: 1479: 1420: 1382: 1366: 1350: 1334: 1326: 1325: 1289: 1273: 1268: 1267: 1261: 1248: 1235: 1229: 1218: 1211: 1204: 1185: 1170: 1160: 1154: 1148: 1134: 1127: 1114: 1108: 1098: 1092: 1077: 1064: 1062: 1060: 1054: 1048: 1041: 1039: 1030: 1024: 1018: 1001: 995: 981: 975: 956: 945: 939: 933: 932:If we exchange 902: 901: 861: 860: 851: 845: 828: 827: 804: 798: 797: 774: 762: 761: 709: 704: 703: 694: 667:still satisfied 647: 641: 635: 624: 613: 607: 604: 589: 573: 562: 516: 486: 470:Évariste Galois 458: 398:Rafael Bombelli 393:complex numbers 316: 306: 267: 248: 243: 237: 165: 105: 58:Évariste Galois 39: 28: 23: 22: 15: 12: 11: 5: 4232: 4230: 4222: 4221: 4211: 4210: 4207: 4206: 4194: 4178: 4177:External links 4175: 4174: 4173: 4164:Modern Algebra 4143: 4137: 4116: 4110: 4097: 4091: 4078: 4072: 4052: 4037: 4020: 4011: 3995: 3978:(4): 645–648, 3972:Amer. J. Math. 3967: 3949: 3931:(7): 357–365. 3917: 3908: 3888: 3857: 3839: 3825: 3801: 3795: 3777: 3774: 3771: 3770: 3763: 3740: 3733: 3707: 3698: 3691: 3673: 3666: 3643: 3636: 3618: 3611: 3593: 3584: 3577: 3548: 3537: 3526: 3519: 3494: 3493: 3491: 3488: 3487: 3486: 3480: 3475: 3469: 3463: 3458: 3450: 3447: 3422: 3419: 3414: 3410: 3389: 3386: 3383: 3380: 3377: 3371: 3368: 3365: 3362: 3359: 3356: 3353: 3350: 3347: 3344: 3341: 3321: 3318: 3315: 3312: 3309: 3304: 3300: 3296: 3293: 3290: 3287: 3264: 3261: 3258: 3255: 3252: 3247: 3243: 3239: 3236: 3233: 3230: 3227: 3224: 3221: 3218: 3213: 3209: 3205: 3202: 3166: 3163: 3160: 3157: 3154: 3149: 3145: 3141: 3138: 3102: 3099: 3089: 2987: 2958: 2920: 2860:Main article: 2857: 2854: 2826: 2686: 2683: 2675: 2653: 2523:solvable group 2518: 2515: 2514: 2513: 2509: 2502:characteristic 2494: 2455: 2430: 2253: 2252: 2203: 2201: 2194: 2188: 2185: 2177: 2176: 2135: 2121: 2070: 2056: 2005: 1954: 1953: 1938: 1935: 1932: 1929: 1926: 1923: 1920: 1917: 1915: 1913: 1910: 1907: 1904: 1901: 1900: 1897: 1891: 1888: 1885: 1882: 1878: 1873: 1870: 1868: 1866: 1863: 1860: 1857: 1854: 1853: 1850: 1844: 1841: 1838: 1835: 1830: 1827: 1821: 1818: 1816: 1814: 1811: 1808: 1805: 1802: 1801: 1781: 1780: 1765: 1762: 1759: 1757: 1755: 1752: 1749: 1746: 1745: 1742: 1739: 1736: 1734: 1732: 1729: 1726: 1725: 1722: 1719: 1716: 1713: 1711: 1709: 1706: 1703: 1702: 1642: 1641: 1626: 1621: 1616: 1611: 1606: 1603: 1600: 1598: 1596: 1593: 1592: 1589: 1584: 1579: 1574: 1569: 1566: 1563: 1561: 1559: 1556: 1555: 1552: 1547: 1542: 1537: 1532: 1529: 1527: 1525: 1522: 1521: 1518: 1513: 1508: 1503: 1498: 1495: 1493: 1491: 1488: 1487: 1469: 1468: 1457: 1454: 1449: 1444: 1441: 1438: 1435: 1432: 1427: 1423: 1419: 1416: 1411: 1406: 1403: 1400: 1397: 1394: 1389: 1385: 1381: 1378: 1373: 1369: 1365: 1362: 1357: 1353: 1349: 1346: 1341: 1337: 1333: 1316: 1315: 1304: 1301: 1296: 1292: 1288: 1285: 1280: 1276: 1260: 1257: 1256: 1255: 1243:If it has two 1241: 986:such that all 930: 929: 918: 915: 912: 909: 895: 894: 883: 880: 877: 874: 871: 868: 842: 841: 826: 821: 816: 813: 810: 807: 805: 803: 800: 799: 796: 791: 786: 783: 780: 777: 775: 773: 770: 769: 751: 750: 739: 736: 733: 730: 727: 724: 721: 716: 712: 693: 690: 626: 625: 576: 574: 567: 561: 558: 529:Camille Jordan 515: 512: 485: 482: 478:solvable group 334:Charles Hutton 326:François Viète 247: 244: 236: 233: 232: 231: 224: 213: 164: 161: 26: 24: 14: 13: 10: 9: 6: 4: 3: 2: 4231: 4220: 4219:Galois theory 4217: 4216: 4214: 4204: 4203:Galois theory 4199: 4195: 4193:at Wiktionary 4192: 4191: 4190:Galois theory 4185: 4181: 4180: 4176: 4172: 4166: 4165: 4158: 4152: 4148: 4144: 4140: 4134: 4130: 4125: 4124: 4117: 4113: 4111:0-387-98541-7 4107: 4103: 4102:Galois Theory 4098: 4094: 4092:0-486-43518-0 4088: 4084: 4079: 4075: 4069: 4065: 4061: 4057: 4053: 4050: 4046: 4040: 4034: 4030: 4026: 4021: 4019: 4014: 4012:0-7167-1480-9 4008: 4004: 4000: 3996: 3993: 3989: 3985: 3981: 3977: 3973: 3968: 3964: 3960: 3959: 3954: 3950: 3946: 3942: 3938: 3934: 3930: 3926: 3922: 3918: 3916: 3911: 3909:0-387-90980-X 3905: 3900: 3899: 3898:Galois Theory 3893: 3889: 3879:on 2008-06-26 3875: 3868: 3867: 3862: 3858: 3854: 3849: 3845: 3840: 3836: 3832: 3828: 3826:0-8218-3817-2 3822: 3818: 3814: 3810: 3806: 3802: 3798: 3796:0-486-62342-4 3792: 3788: 3787:Galois Theory 3784: 3780: 3779: 3775: 3766: 3764:9780387942254 3760: 3756: 3755: 3750: 3744: 3741: 3736: 3730: 3726: 3722: 3718: 3711: 3708: 3702: 3699: 3694: 3688: 3684: 3677: 3674: 3669: 3667:9783528084981 3663: 3656: 3655: 3647: 3644: 3639: 3633: 3629: 3622: 3619: 3614: 3608: 3604: 3597: 3594: 3588: 3585: 3580: 3574: 3570: 3565: 3564: 3558: 3552: 3549: 3546: 3541: 3538: 3535: 3530: 3527: 3522: 3520:0-412-34550-1 3516: 3512: 3511:Galois Theory 3508: 3502: 3500: 3496: 3489: 3484: 3481: 3479: 3476: 3473: 3470: 3467: 3464: 3462: 3459: 3456: 3453: 3452: 3448: 3446: 3444: 3440: 3436: 3420: 3417: 3412: 3408: 3384: 3381: 3378: 3369: 3366: 3360: 3354: 3351: 3348: 3345: 3342: 3316: 3313: 3310: 3302: 3298: 3294: 3291: 3288: 3285: 3278: 3259: 3256: 3253: 3245: 3241: 3237: 3234: 3231: 3225: 3222: 3219: 3211: 3207: 3203: 3200: 3192: 3188: 3184: 3180: 3161: 3158: 3155: 3147: 3143: 3139: 3136: 3128: 3124: 3120: 3116: 3112: 3108: 3100: 3098: 3096: 3095:Monster group 3088: 3084: 3083:Mathieu group 3080: 3079:simple groups 3075: 3070: 3065: 3059: 3056: 3050: 3046: 3041: 3036: 3031: 3026: 3020: 3014: 3008: 3002: 2998: 2990: 2986: 2980: 2977:of symmetric 2975: 2971:is the field 2969: 2961: 2957: 2953: 2949: 2943: 2937: 2931: 2923: 2919: 2912: 2906: 2902: 2897: 2892: 2887: 2881: 2876: 2871: 2869: 2863: 2855: 2853: 2851: 2847: 2842: 2838: 2834: 2825: 2818: 2814: 2810: 2805: 2801: 2797: 2793: 2789: 2781: 2777: 2770: 2766: 2760: 2757: 2755: 2750: 2746: 2742: 2738: 2734: 2729: 2722: 2716: 2709: 2705: 2701: 2697: 2691: 2684: 2682: 2678: 2674: 2670: 2667:, namely the 2666: 2663:, noncyclic, 2662: 2656: 2652: 2648: 2642: 2637: 2633: 2628: 2621: 2615: 2612: 2607: 2602: 2599: 2593: 2587: 2582: 2579: 2571: 2565: 2561: 2555: 2550: 2546: 2542: 2537: 2533: 2528: 2524: 2516: 2510: 2507: 2503: 2499: 2495: 2491: 2486: 2481: 2475: 2471: 2465: 2460: 2456: 2453: 2449: 2448:finite fields 2445: 2444:number fields 2441: 2436: 2431: 2428: 2424: 2423: 2422: 2419: 2416: 2412: 2408: 2404: 2400: 2396: 2383:by adjoining 2381: 2367: 2362: 2357: 2351: 2340: 2334: 2331: 2327: 2321: 2315: 2309: 2303: 2301: 2300:Galois groups 2296: 2290: 2285: 2284:automorphisms 2280: 2274: 2268: 2264: 2260: 2249: 2246: 2238: 2228: 2224: 2220: 2214: 2213: 2209: 2204:This section 2202: 2198: 2193: 2192: 2186: 2184: 2182: 2169: 2165: 2161: 2157: 2153: 2149: 2145: 2141: 2136: 2119: 2104: 2100: 2096: 2092: 2088: 2084: 2080: 2076: 2071: 2054: 2039: 2035: 2031: 2027: 2023: 2019: 2015: 2011: 2006: 1998: 1994: 1990: 1986: 1982: 1978: 1974: 1970: 1965: 1964: 1963: 1960: 1936: 1930: 1924: 1921: 1918: 1916: 1908: 1902: 1895: 1886: 1880: 1876: 1871: 1869: 1861: 1855: 1848: 1839: 1833: 1828: 1825: 1819: 1817: 1809: 1803: 1792: 1791: 1790: 1787: 1763: 1760: 1758: 1753: 1750: 1747: 1740: 1737: 1735: 1730: 1727: 1720: 1717: 1714: 1712: 1707: 1704: 1693: 1692: 1691: 1688: 1685: 1679: 1673: 1667: 1661: 1653: 1651: 1647: 1624: 1619: 1614: 1609: 1604: 1601: 1599: 1594: 1587: 1582: 1577: 1572: 1567: 1564: 1562: 1557: 1550: 1545: 1540: 1535: 1530: 1528: 1523: 1516: 1511: 1506: 1501: 1496: 1494: 1489: 1478: 1477: 1476: 1474: 1455: 1447: 1442: 1439: 1436: 1433: 1430: 1425: 1421: 1409: 1404: 1401: 1398: 1395: 1392: 1387: 1383: 1376: 1371: 1367: 1363: 1360: 1355: 1347: 1344: 1339: 1335: 1324: 1323: 1322: 1320: 1302: 1299: 1294: 1290: 1286: 1283: 1278: 1274: 1266: 1265: 1264: 1258: 1251: 1246: 1242: 1238: 1232: 1225: 1221: 1214: 1207: 1200: 1196: 1192: 1188: 1181: 1177: 1173: 1168: 1167: 1166: 1163: 1157: 1151: 1145: 1141: 1137: 1131: 1126: 1122: 1117: 1111: 1106: 1105:transposition 1101: 1095: 1090: 1084: 1080: 1074: 1072: 1057: 1051: 1037: 1033: 1027: 1021: 1015: 1013: 1009: 1004: 998: 993: 989: 984: 978: 973: 969: 963: 959: 952: 948: 942: 936: 916: 913: 910: 907: 900: 899: 898: 881: 878: 875: 872: 869: 866: 859: 858: 857: 854: 848: 824: 819: 814: 811: 808: 806: 801: 794: 789: 784: 781: 778: 776: 771: 760: 759: 758: 756: 753:By using the 737: 734: 731: 728: 725: 722: 719: 714: 710: 702: 701: 700: 699: 696:Consider the 691: 689: 687: 683: 678: 676: 672: 668: 664: 660: 654: 650: 644: 638: 633: 622: 619: 611: 601: 597: 593: 587: 586: 582: 577:This section 575: 571: 566: 565: 559: 557: 555: 551: 547: 543: 538: 534: 530: 526: 522: 513: 511: 508: 504: 499: 490: 481: 479: 475: 471: 467: 461: 456: 452: 447: 445: 441: 437: 436: 430: 429:Paolo Ruffini 425: 423: 418: 414: 410: 406: 401: 399: 394: 390: 386: 382: 380: 375: 371: 367: 363: 358: 357:for details. 356: 351: 345: 341: 339: 338:Albert Girard 335: 331: 327: 322: 319: 313: 309: 303: 299: 295: 291: 287: 283: 279: 275: 271: 265: 261: 257: 253: 245: 242: 234: 229: 225: 222: 218: 214: 211: 210:constructible 207: 203: 202: 201: 199: 195: 190: 188: 183: 179: 173: 168: 162: 160: 158: 154: 149: 147: 142: 140: 136: 135:constructible 132: 128: 124: 119: 117: 113: 109: 102: 98: 94: 90: 86: 82: 78: 73: 71: 67: 63: 59: 55: 54:Galois theory 51: 42: 37: 32: 19: 4189: 4170: 4163: 4156: 4150: 4122: 4101: 4082: 4059: 4045:Grothendieck 4024: 4017: 4002: 3975: 3971: 3956: 3928: 3924: 3914: 3897: 3881:. Retrieved 3874:the original 3865: 3843: 3808: 3786: 3753: 3743: 3716: 3710: 3701: 3682: 3676: 3653: 3646: 3627: 3621: 3602: 3596: 3587: 3562: 3551: 3545:Cardano 1545 3540: 3529: 3510: 3507:Stewart, Ian 3455:Galois group 3193:is assigned 3190: 3186: 3178: 3123:vector space 3118: 3114: 3104: 3086: 3073: 3063: 3060: 3054: 3048: 3044: 3034: 3024: 3018: 3012: 3006: 3000: 2996: 2988: 2984: 2973: 2967: 2959: 2955: 2951: 2947: 2941: 2935: 2929: 2921: 2917: 2910: 2904: 2895: 2885: 2879: 2875:ground field 2872: 2867: 2865: 2843: 2836: 2832: 2823: 2819: 2812: 2808: 2806: 2799: 2795: 2791: 2787: 2779: 2775: 2768: 2764: 2761: 2758: 2751: 2744: 2740: 2736: 2732: 2726: 2714: 2707: 2703: 2699: 2695: 2676: 2672: 2654: 2650: 2640: 2626: 2619: 2616: 2610: 2605: 2603: 2597: 2591: 2585: 2577: 2569: 2563: 2559: 2553: 2541:factor group 2535: 2531: 2527:group theory 2520: 2512:polynomials. 2489: 2479: 2473: 2469: 2463: 2452:local fields 2434: 2420: 2414: 2410: 2406: 2402: 2398: 2394: 2379: 2365: 2355: 2349: 2338: 2335: 2329: 2325: 2319: 2313: 2307: 2304: 2294: 2288: 2278: 2272: 2266: 2262: 2256: 2241: 2232: 2217:Please help 2205: 2178: 2167: 2163: 2159: 2155: 2151: 2147: 2143: 2139: 2102: 2098: 2094: 2090: 2086: 2082: 2078: 2074: 2037: 2033: 2029: 2025: 2021: 2017: 2013: 2009: 1996: 1992: 1988: 1984: 1980: 1976: 1972: 1968: 1958: 1955: 1785: 1782: 1689: 1683: 1677: 1671: 1665: 1662: 1654: 1646:permutations 1643: 1470: 1317: 1262: 1249: 1240:are swapped. 1236: 1230: 1223: 1219: 1212: 1205: 1198: 1194: 1190: 1186: 1179: 1175: 1171: 1161: 1155: 1149: 1143: 1139: 1135: 1132: 1115: 1109: 1099: 1093: 1082: 1078: 1075: 1071:not rational 1055: 1049: 1035: 1031: 1025: 1019: 1016: 1002: 996: 988:coefficients 982: 976: 967: 961: 957: 950: 946: 940: 934: 931: 896: 852: 846: 843: 752: 695: 686:Galois group 679: 666: 662: 659:permutations 652: 648: 642: 636: 629: 614: 605: 590:Please help 578: 549: 532: 524: 517: 502: 495: 474:Galois group 465: 459: 454: 450: 448: 434: 433:permutation 426: 421: 417:permutations 416: 404: 402: 388: 383:His student 377: 359: 350:discriminant 347: 343: 323: 317: 311: 307: 301: 297: 293: 289: 285: 281: 277: 273: 269: 249: 191: 181: 175: 170: 166: 150: 143: 120: 96: 88: 74: 66:group theory 62:field theory 53: 47: 40: 4056:Lang, Serge 3866:Artis Magnæ 3783:Artin, Emil 3749:Lang, Serge 3717:Polynomials 3127:derivations 3030:fixed field 2717:= 1.1673... 2659:contains a 546:Eugen Netto 422:composition 305:, where 1, 246:Pre-history 81:polynomials 50:mathematics 3883:2015-01-10 3853:2010.15707 3776:References 2893:says that 2850:Emil Artin 2846:Serge Lang 2575:primitive 2567:the field 2004:(identity) 1245:irrational 1121:isomorphic 455:particular 389:Ars Magna. 262:sign) the 239:See also: 4049:groupoids 3963:EMS Press 3835:118256821 3789:. Dover. 3785:(1998) . 3571:–3, 302. 3418:⊂ 3382:∈ 3376:∀ 3346:∈ 3289:⊂ 3232:⊂ 3028:. 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