On the Galois module structure over CM-fields
For a given number field \(K\) and a given finite abelian group \(\Delta\), let us call \(R(K,\Delta)\) the set of all realizations of \(\Delta\) as a Galois group \(\text{Gal}(N/K)\) of a tame field extension \(N\) of \(K\). Question: to what extent are the elements of \(R(K,\Delta)\) characterized by their ``ramification'' and their ``Galois module structure''? The main result of this paper is that, for a given CM-field \(K\), the unramified realizations in \(R(K,\Delta)\) are invariant under the involution of \(R(K,\Delta)\) which is induced by complex conjugation on \(K\) and by inversion on \(\Delta\). Several applications are given. For example, if \(\Delta\) is of odd order and \(K\) is totally real, the answer to the above question is affirmative.
scientific article; zbMATH DE number 91467
scientific article