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
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