Swan modules and realisable classes for Kummer extensions of prime degree
If one fixes a cyclic group of prime order \(G\) and a number field \(K\), then the set of classes \(R\) of \({\mathbb Z} G\)-modules realisable as \(O_L\) with \(L\) a tame \(G\)-extension of \(K\) forms a subgroup \(R\) of the class group of \({\mathbb Z} G\). McCulloh gave a very explicit description of \(R\) in terms of a Stickelberger ideal. In the present paper, the author proves a result bounding \(R\) from below, based on McCulloh's description. More precisely he shows that if \(K\) contains \(\zeta_l\), then \(R\) contains \(T^{(l-1)/2}\), where \(T\) is the so-called Swan subgroup. The second main result of the paper (which, incidentally, does not use the first one) gives criteria just when \(O_L\) is in \(T\), following an earlier result by Gómez Ayala. Finally, as a consequence of the first theorem, the author deduces that \(R\) is nontrivial if \(K=\mathbb{Q}(\zeta_l)\) with \(l>3\) any prime. A more general result in this direction is proved in a forthcoming joint paper of the author, \textit{K. Rubin, A. Srivastav}, and the reviewer [J. Number Theory (to appear)] for every number field \(K\) except \({\mathbb Q}\), one may find a prime \(l\) (infinitely many in fact) such that \(R\) is nontrivial.
scientific article; zbMATH DE number 1271818