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Extensions régulières de \({\mathbb{Q}}(T)\) de groupe de Galois \(\tilde A_ n\). (Regular extensions of \({\mathbb{Q}}(T)\) with Galois group \(\tilde A_ n)\). (French) Zbl 0714.11074
Let \(\tilde A_ n\) be the double cover of the alternating group \(A_ n\), \(n\geq 4\). In this paper it is proved that \(\tilde A_ n\) appears as Galois group of a regular extension of \({\mathbb{Q}}(T)\), for all \(n\geq 4\). Consequently, \(\tilde A_ n\) is Galois group over every number field, for all \(n\geq 4\). In order to obtain the result in the odd case, the author considers polynomials of the type \(F_ T(X)=P(X)-TQ(X)\in {\mathbb{Q}}(T)[X].\) He gives explicit conditions to have that the inertial group of any ramified prime of \({\mathbb{Q}}(T)\) in a splitting field of \(F_ T(X)\) is generated by a 3-cycle and that the discriminant of \(F_ T(X)\) is a square. Therefore, the Galois group of \(F_ T(X)\) over \({\mathbb{Q}}(T)\) is \(A_ n\). On the other hand a recent result of Serre assures that, since the inertial group of any prime has odd order, the quadratic form \(Tr_{E/{\mathbb{Q}}(T)}(X^ 2)\) is equivalent over \({\mathbb{Q}}(T)\) to a form with coefficients in \({\mathbb{Q}}\), where \(E={\mathbb{Q}}(T)[X]/(F_ T(X))\) (cf. App. 2 and App. 3). The author proves that the polynomial P(X) can be chosen in such a way that the trace form associated to P(X) is the trivial one. Hence the obstruction to the embedding problem in \(\tilde A_ n\) associated to \(F_ T(X)\) is trivial [see J.-P. Serre, Comment. Math. Helv. 59, 651-676 (1984; Zbl 0565.12014)]. The theorem in the even case is obtained from the odd one. In the case \(n=7\) a polynomial \(F_ T(X)\) is explicitly given.
Reviewer: N.Vila

11R32 Galois theory
12F12 Inverse Galois theory
11R58 Arithmetic theory of algebraic function fields
Full Text: DOI
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