Explicit construction of \(\tilde A_ n\) type fields
Let \(\tilde G\) be the preimage of a subgroup G of the alternating group \(A_ n\), \(n\geq 4\), in the double cover \(\tilde A_ n\) of \(A_ n\). Let K be a field of characteristic \(\neq 2\), and \(L/K\) be a Galois extension with Galois group isomorphic to G. The goal of this paper is to construct explicit solutions to the embedding problem associated to \(\tilde G\to G\simeq Gal(L/K)\), when this embedding problem is solvable. The author proves that any non-zero coordinate \(\gamma\) of the spin norm of an explicit element in the Clifford algebra of a quadratic form trace associated to \(L/K\) provides a solution to the solvable embedding problem. The general solution is then \(L((r\gamma)^{1/2})\), where r runs through \(K^ x/K^{x2}\). Moreover, in some generic special cases, this element is computed in terms of matrices attached to the quadratic form. In particular, it is checked that this construction recovers a result of \textit{E. Witt} [J. Reine Angew. Math. 174, 237-245 (1936; Zbl 0013.19601)]. This answers a question raised by \textit{J.-P. Serre} in [Comment. Math. Helv. 59, 651-676 (1984; Zbl 0565.12014)].
scientific article; zbMATH DE number 4154557