Stably free rings of integers over \(\mathbb{Z}[H_8\times C_2]\)
Let \(G= H_8\times C_2\), the direct product of the quaternion group of order 8 with a cyclic group of order 2. It is known that there are four isomorphism classes of non-free stably free \(\mathbb{Z} G\)-modules. The aim of this paper is to show that each of these four classes is represented infinitely often by rings of integers \({\mathcal O}_N\), where \(N\) is a tamely ramified Galois extension of \(\mathbb{Q}\) with Galois group \(G\). This bald statement of the result hides the immense amount of calculation that is needed; these calculations will be of independent interest to anyone who whishes to compute explicitly with quaternionic and related extensions of the rational numbers. Sections I to V detail the method of constructing quaternionic and \(G\)-extensions of \(\mathbb{Q}\), together with an analysis of the ambiguous ideals in such extensions and their quadratic and biquadratic subfields. The methods are adaptions of those used by Witt and Martinet. Section VI reviews the computation of the locally free class group of \(\mathbb{Q} G\), and identifies the isomorphism classes of non-free stably free \(\mathbb{Z} G\)-modules in terms of units of the local ring \(\mathbb{F}_2 H_8\), thus recovering results due to Swan. These ingredients are combined in section VII to produce explicit infinite families of extensions, as promised. Finally, the author considers a numerical example that contains the field \(\mathbb{Q} (\sqrt{1001}, \sqrt{2805})\).
representations by rings of algebraic integers
scientific article; zbMATH DE number 1229653
scientific article