Caputo, Luca; Cobbe, Alessandro An explicit candidate for the set of Steinitz classes of tame Galois extensions with fixed Galois group of odd order. (English) Zbl 1292.11127 Proc. Lond. Math. Soc. (3) 107, No. 2, 391-413 (2013). 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 L. R. McCulloh since 2002 (see “From Galois module classes to Steinitz classes”, Preprint, 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 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 [A. Cobbe, J. Number Theory 130, No. 5, 1129–1154 (2010; Zbl 1215.11108)]. Reviewer: Bouchaïb Sodaïgui (Valenciennes) Cited in 1 Document MSC: 11R33 Integral representations related to algebraic numbers; Galois module structure of rings of integers 11R04 Algebraic numbers; rings of algebraic integers Keywords:ring of integers; Steinitz class; class group; discriminant; Galois group Citations:Zbl 1215.11108 Software:PARI/GP PDFBibTeX XMLCite \textit{L. Caputo} and \textit{A. Cobbe}, Proc. Lond. Math. Soc. (3) 107, No. 2, 391--413 (2013; Zbl 1292.11127) Full Text: DOI arXiv