On realizable Galois module classes and Steinitz classes of nonabelian extensions
To a tame \(\Gamma\)-Galois extension \(N/k\) of number fields, one can attach a class in \(cl({\mathcal M})\) (the Galois module class) and also a class in \(cl(O_k)\) (the Steinitz class). Here \(\mathcal M\) is a maximal order in \(k[\Gamma]\) containing \(O_k[\Gamma]\). (The Galois module class can actually be defined in the considerably larger class group of \(O_k[\Gamma]\) itself.) The standard conjectures, as yet unproved for general \(\Gamma\), say that the subset of all Galois module classes (of all Steinitz classes, respectively) with fixed data \(k\) and \(\Gamma\) is a \textit{subgroup} of \(cl({\mathcal M})\) (\(cl(O_k)\)). The article under review resumes the setting of a paper of \textit{N. P. Byott, B. Sodaïgui}, and the reviewer [[BGS]: ``Classes réalisables d'extensions non-abeliennes'', J. Reine Angew. Math. 601, 1--27 (2006; Zbl 1137.11069)] and generalizes its results. More precisely: [BGS] considered groups \(\Gamma\) that arise as semidirect products of a 2-elementary group \(V\) of order \(2^r\) and a cyclic group \(C\) of order \(2^r-1\) that acts faithfully on \(V\). It is proved that the standard conjectures hold for this particular group. The paper under review replaces 2 by any prime, more precisely: \(V\) is now a finite vector space over the field of \(p\) elements, and \(|V|=p^r\) say. The other conditions are adjusted accordingly (\(C\) has order \(p^r-1\) and again acts faithfully on \(V\)). The main results say that the standard conjectures are again true for the new group \(\Gamma\), provided that \(k\) contains a primitive \(p\)-th root of unity. (Note that the latter condition is void for \(p=2\).) To a large extent, the arguments run parallel to [BGS]; this is clearly indicated by the authors. (In particular \(\zeta_p\in k\) is required in order to apply Kummer theory to \(V\)-extensions, as before.) But there are a few notable differences, which are again well explained; for instance a central argument concerning weights of words in a particular class of cyclic codes has to be done anew. The nice link to coding theory that came up in [BGS] continues to be important. As the authors say themselves, the paper gives a lot of details, including many details taken from [BGS], for the reader's convenience. While one of the justifications the authors give for this (``non-francophone readers''; note [BGS] is in French, and the paper being reviewed is in English) is perhaps not extremely cogent, the authors' strategy certainly has resulted in a very readable and well-written paper.
scientific article; zbMATH DE number 5378839
scientific article