An explicit candidate for the set of Steinitz classes of tame Galois extensions with fixed Galois group of odd order (Q2843975)

Let \(G\) be a finite group. Let \(k\) be a number field and \(\mathrm{Cl}(k)\) its class group. Let \(R_t(k, G)\) be the set of Steinitz classes of tamely ramified \(G\)-Galois extensions of \(k\). There is a conjecture stating that \(R_t(k, G)\) is a subgroup of \(\mathrm{Cl}(k)\). The authors first define an explicit subgroup (which is the explicit candidate in the title) \(\mathcal{W}(k, G)\) of \(\mathrm{Cl}(k)\) and then prove that \(R_t(k, G)\subset \mathcal{W}(k, G)\). The reviewer points out that the group \(\mathcal{W}(k, G)\) and the inclusion \(R_t(k, G)\subset \mathcal{W}(k, G)\) have been well-known by the work of \textit{L. R. McCulloh} since 2002 (see ``From Galois module classes to Steinitz classes'', Preprint, \url{arxiv:1207.5702v1}). The novelty of the paper is the following. The authors prove that \(R_t(k, G)=\mathcal{W}(k, G)\) in the particular case where \(G\) has order dividing \(l^4\), \(l\) is an odd prime number (in the case where is \(G\) abelian, this is well-known by a result of \textit{L. P. Endo} [``Steinitz classes of tamely ramified Galois extensions of algebraic number fields'', PhD thesis, University of Illinois at Urbana-Champaign (1975)] ; if \(G\) is non abelian of order \(l^3\), it is also known by a combination of results from several authors). The proof uses a refinement of techniques of the second author introduced in [\textit{A. Cobbe}, J. Number Theory 130, No. 5, 1129--1154 (2010; Zbl 1215.11108)].