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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

##### MSC:
 11R32 Galois theory 12F12 Inverse Galois theory 11R58 Arithmetic theory of algebraic function fields
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##### References:
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