\(\tilde A_5\) and \(\tilde A_7\) are Galois groups over number fields
This paper deals with the problem of the realizability of \(\tilde A_n\), the double cover of the alternating group \(A_n\), as Galois group over number fields. This question can be regarded as an embedding problem whose obstruction is related to the Witt invariant of an associated quadratic trace form, by means of \textit{J.-P. Serre}'s trace formula [Comment. Math. Helv. 59, 651--676 (1984; Zbl 0565.12014)]. The author considers generalized Laguerre polynomials \(F_n(X,\lambda,\mu)\in \mathbb Q(\lambda,\mu)[X]\) (cf. [\textit{I. Schur}, Collected Works, Vol. III (1973; Zbl 0274.01054)]) and computes the principal minors of the associated trace form matrix. Then, he shows that for \(n\equiv 3\pmod 4\) the obstruction to the embedding problem in \(\tilde A_ n\) associated to the realization of \(A_n\) given by \(F_n(X,1,1)\in\mathbb Q[X]\) is trivial and, therefore, that for \(n\equiv 3\pmod 4\) \(\tilde A_n\) appears as Galois group over \(\mathbb Q\). The study of suitable Diophantine equations allows the author to prove that there exist infinitely many realizations of \(A_5\) and of \(A_7\) over \(\mathbb Q\) given by Laguerre polynomials which can be embedded in \(\tilde A_5\) and \(\tilde A_7\), respectively. Therefore, \(\tilde A_5\) and \(\tilde A_7\) appear as Galois group over every number field. Let \(K\) be a number field, a finite group \(G\) is called \(K\)-admissible if there exists a Galois extension of \(K\) with Galois group \(G\) which is a maximal subfield of a finite-dimensional division algebra with center \(K\). The author shows that if \(\sqrt{10}\not\in \widehat K(\sqrt{15})\), \(A_5\) is \(K\)-admissible, and if \(\sqrt{10}\not\in \widehat K(\sqrt{3},\sqrt{5},\sqrt{- 2})\), \(\tilde A_5\) is \(K\)-admissible, where \(\widehat K\) denotes the Galois closure of \(K\) over \(\mathbb Q\).
scientific article; zbMATH DE number 3985352
scientific article