Relative Galois structure of rings of integers.
Given a number field \(k\) and a finite abelian group \(\Gamma\); let \(\pi: \Gamma\simeq \text{Gal}(N/k)\) run over all realizations of \(\Gamma\) as Galois group over \(k\) with \(N/k\) at most tamely ramified. The problem of the realizable classes of \((k,\Gamma)\) asks to determine the subset of the class group of the group ring \(O_ k\Gamma\) of \(\Gamma\) over the ring of integers \(O_ k\) which are realized as \((O_ N)_{O_ k}\) where \(\pi\) runs as above; it is also of interest to impose additional `embedding problem' conditions on \(\pi\) [see for example p. 222--231 of the book by \textit{A. Fröhlich}, ``Galois module structure of algebraic integers.'' Berlin etc.: Springer (1983; Zbl 0501.12012) for an account of this idea]. We refer to the papers by \textit{L. R. McCulloh} (the most recent one appeared in [J. Reine Angew. Math. 375--376, 259--306 (1987; Zbl 0619.12008)]) for precise results on the realizable classes in the general situation. The paper under review is for a great part expository; some new results in special cases are given: For example, let \([k:{\mathbb Q}]=2\), \(\Gamma\simeq {\mathbb Z}/\ell\), \(\ell\) a prime number, and suppose that one considers the image of the realizable classes under the natural homomorphism \(Cl(O_ k\Gamma)\to Cl({\mathfrak M})\) where \({\mathfrak M}\) is the maximal order in \(k\Gamma\) ; the result is that this image does not become smaller if one imposes upon \(N\) the extra condition that the extension \(N/{\mathbb Q}\) is not normal.
scientific article; zbMATH DE number 4043961
scientific article