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On central extensions of \(A_ n\) as Galois group over \({\mathbb{Q}}\). (English) Zbl 0562.12011
This paper contains a result which shows that every central extension of the alternating group \(A_ n\) occurs as a Galois group over \({\mathbb{Q}}\) for infinitely many n, and furthermore, that every central extension of \(A_ n\) occurs as a Galois group over \({\mathbb{Q}}(\sqrt{-1})\) if \(n\neq 6,7\). The author constructs new families of equations with Galois group \(A_ n\) over \({\mathbb{Q}}(T)\) and solves a central embedding problem for \(A_ n\cong Gal(K/{\mathbb{Q}}))\) by a recent method of Serre which connects the obstruction to the solvability of this embedding problem with the Hasse-Witt-invariant of the trace form of K/\({\mathbb{Q}}\) [see J.-P. Serre, Comment. Math. Helv. 59, 651-676 (1984)].
Reviewer: H.Opolka

11R32 Galois theory
20F29 Representations of groups as automorphism groups of algebraic systems
11E16 General binary quadratic forms
20B25 Finite automorphism groups of algebraic, geometric, or combinatorial structures
Full Text: DOI
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