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QUARTIC EXERCISES It Is Known Since the Xvith Century That the Solution of Quartic Equations Can Be Obtained by Means of Auxilia

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QUARTIC EXERCISES It Is Known Since the Xvith Century That the Solution of Quartic Equations Can Be Obtained by Means of Auxilia QUARTIC EXERCISES MAX-ALBERT KNUS AND JEAN-PIERRE TIGNOL Abstract. A correspondence between quartic ¶etale algebras over a ¯eld and quadratic ¶etale extensions of cubic ¶etale algebras is set up and investigated. The basic constructions are laid out in general for sets with a pro¯nite group action and for torsors, and translated in terms of ¶etale algebras and Galois algebras. In the ¯nal section, a parametrization of cyclic quartic algebras is given. It is known since the XVIth century that the solution of quartic equations can be obtained by means of auxiliary equations of degree 3, called cubic resolvents. The situation is easily understood in terms of Galois theory. For any integer n 1, let ¸ Sn denote the symmetric group on 1; : : : ; n . The symmetric group S4 contains a normal subgroup of order 4, Klein'sf Viererggruppe V = I; (1; 2)(3; 4); (1; 3)(2; 4); (1; 4)(2; 3) ; f g which is the kernel of the action of S4 on its three Sylow 2-subgroups. Numbering from 1 to 3 these Sylow subgroups, we get an exact sequence of groups ½ (1) 1 V S S 1: ! ! 4 ¡! 3 ! Let F be an arbitrary ¯eld and P F [X] be a separable polynomial of degree 4. Let also F be a separable closure 2of F and Q F be the sub¯eld generated by s ½ s the roots of P . The Galois group Gal(Q=F ) can be viewed as a subgroup of S4 through its action on the roots of P . The sub¯eld L of Q ¯xed under Gal(Q=F ) V is generated by the roots of a cubic resolvent, as was shown by Lagrange. Fo\r a given quartic polynomial P , there are actually many polynomials of degree 3 which qualify as cubic resolvents; only the extension L=F is an invariant of P (or of Q). Galois cohomology provides another viewpoint on this construction. Since Sn is the automorphism group of the ¶etale F -algebra F n = F F , it is well-known 1 £ ¢ ¢ ¢ £ that the Galois cohomology set H (F; Sn) is in canonical one-to-one correspondence with the isomorphism classes of ¶etale F -algebras of degree n, see [3, (29.9)]. The map ½ in (1) induces a map ½1 : H1(F; S ) H1(F; S ) 4 ! 3 which associates to every quartic ¶etale F -algebra Q a cubic ¶etale F -algebra (Q) uniquely determined up to isomorphism. If P F [X] is a separable polynomRial of degree 4 with cubic resolvent R, and if Q is the2 factor algebra Q = F [X]=P , then (Q) F [X]=R, see 4.3. R Our'¯rst aim is toxmake explicit the construction of (Q) from Q. But this R construction can be further extended. Each of the three Sylow 2-subgroups of S4 The authors gratefully acknowledge support from the RT Network \K-theory, Linear Algebraic Groups and Related Structures" (contract HPRN-CT-2002-00287). The ¯rst author is partially supported by the Swiss National Foundation under grant 21-064712.01. The second author is partially supported by the National Fund for Scienti¯c Research (Belgium). 1 2 MAX-ALBERT KNUS AND JEAN-PIERRE TIGNOL contains two transpositions, and each transposition is in one and only one Sylow subgroup, hence the set of transpositions can be viewed as a double covering of the set of Sylow 2-subgroups. Therefore, the conjugation action of S4 on its six transpositions de¯nes a map ¸: S S S ; 4 ! 2 o 3 where the wreath product S2 S3 is viewed as the group of automorphisms of a double covering of a set of throee elements (see 3.1). The map ¸ extends to an isomorphism of groups x » ¸^ : S2 S4 S2 S3; 1 £ ! o see 4.2. The set H (F; S2 S3) classi¯es the quadratic ¶etale extensions of cubic ¶etalex F -algebras (see 3.2), aond the induced bijection x ¸^1 : H1(F; S ) H1(F; S ) » H1(F; S S ) 2 £ 4 ! 2 o 3 associates to every pair consisting of a quartic ¶etale F -algebra Q and a quadratic ¶etale F -algebra a quadratic ¶etale extension of the cubic resolvent (Q). In 4.3, we give an explicit construction of this quadratic extension, and we descrR ibe in x4.4 the x inverse of ¸^1, attaching a quartic ¶etale F -algebra and a quadratic ¶etale F -algebra to any quadratic ¶etale extension of a cubic ¶etale F -algebra. In the ¯nal sections, we classify quartic ¶etale algebras and their associated qua- dratic extensions of cubic algebras according to their decomposition into direct products of ¯elds (see 5.3) and we parametrize cyclic quartic extensions. x Contents 1. ¡-sets and coverings 3 1.1. Basic constructions on ¡-sets 3 1.2. Coverings 4 2. Eta¶ le algebras and extensions 7 2.1. Basic constructions on ¶etale algebras 8 2.2. Extensions of ¶etale algebras 11 3. Cohomology of permutation groups 14 3.1. Permutations 14 3.2. Cohomology and ¡-sets 14 3.3. Torsors 16 3.4. Cohomology and ¶etale algebras 17 3.5. Galois algebras 18 4. The symmetric group on four elements 20 4.1. Sets of four elements and double coverings 21 4.2. Cohomology 24 4.3. Quartic ¶etale algebras 29 4.4. Quadratic extensions of cubic ¶etale algebras 31 5. Special actions on four elements 36 5.1. Action through S3 38 5.2. Action through D4 38 5.3. Classi¯cation of quartic algebras 43 6. Cyclic quartic algebras 43 6.1. Characteristic not 2 44 6.2. Characteristic 2 47 QUARTIC EXERCISES 3 References 49 La recherche des extensions d'un corps k dont le groupe de Ga- lois sur k est S4 ou A4 n'est pas autre chose, du point de vue des alg¶ebristes du XIXe si`ecle, que la th¶eorie de l'¶equation du 4e degr¶e. C'est un probl`eme pour lequel ces alg¶ebristes n'avaient que du m¶epris. (A. Weil) 1. ¡-sets and coverings 1.1. Basic constructions on ¡-sets. Let ¡ be a pro¯nite group, which will be ¯xed throughout this section. Finite sets with a continuous action of ¡ are called ¡-sets. We let X denote the number of elements in a ¡-set X. If X is a ¡-set with n elements,j ajnd k is a positive integer, k n, we let § (X) denote the set of · k k-tuples of pairwise distinct elements of X and ¤k(X) the set of k-element subsets of X; thus § (X) = (» ; : : : ; » ) Xk » = » for i = j ; k 1 k 2 j i 6 j 6 ¤ (X) = » ; : : : ; » X » = » for i = j : k ©f 1 kg ½ j i 6 j 6 ª The action of ¡ on X induces© actions on §k(X) and ¤k(X),ªhence §k(X) and ¤k(X) are ¡-sets, and we have n n § (X) = k! ; ¤ (X) = : j k j k j k j k µ ¶ µ ¶ The symmetric group Sk acts on §k(X) by permutation of the entries, and we may consider ¤k(X) as the set of orbits of §k(X) under this action, i.e. as the quotient ¡-set ¤k(X) = §k(X)=Sk: For k = n, we may also consider the action of the alternating group An on §n(X). The quotient is called the discriminant of X and denoted by ¢(X), ¢(X) = §n(X)=An; see [3, p. 291]. This is a ¡-set with ¢(X) = 2 if n 2. If n is even, n = 2m, let j j ¸ γ : ¤ (X) ¤ (X) X m ! m be the map which associates to every m-element subset of X its complementary 2 subset. Since γX = Id, this map de¯nes an action of S2 on ¤m(X). The map γX is ¡-equivariant (i.e. compatible with the action of ¡), hence the quotient (X) = ¤ (X)=S R m 2 is a ¡-set. It is the set of partitions of X into m-element subsets. Example 1.1. If X = 1; 2; 3; 4 , then f g ¤ (X) = 1; 2 ; 3; 4 ; 1; 3 ; 2; 4 ; 1; 4 ; 2; 3 2 f g f g f g f g f g f g and © ª (X) = 1; 2 ; 3; 4 ; 1; 3 ; 2; 4 ; 1; 4 ; 2; 3 : R f g f g f g f g f g f g n© ª © ª © ªo 4 MAX-ALBERT KNUS AND JEAN-PIERRE TIGNOL If X = 2, the map j j γ : X = ¤ (X) ¤ (X) = X X 1 ! 1 interchanges the two elements of X. For X, X 0 two ¡-sets with X = X0 = 2, the map j j j j 0 0 γ γ 0 : X X X X X £ X £ ! £ de¯nes an action of S2 compatible with the ¡-action. Let X X0 = (X X0)=S ; ¤ £ 2 a ¡-set with X X0 = 2. Thus, if X = x ; x and X0 = x0 ; x0 , then j ¤ j f 1 2g f 1 2g X X0 = (x ; x0 ); (x ; x0 ) ; (x ; x0 ); (x ; x0 ) : ¤ f 1 1 2 2 g f 1 2 2 1 g The following observations©are clear: ª Proposition 1.2. Let X, X 0, be ¡-sets of two elements. (a) The ¡-action on X X is trivial. (b) If the ¡-action on X¤0 is trivial, then X X0 X. (Note that the isomor- phism is not canonical.) ¤ ' (c) The operation de¯nes a group structure on the set of isomorphism classes of ¡-sets of two¤ elements. See 3.2 for a cohomological interpretation of the group structure induced by .
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