Non-existence and splitting theorems for normal integral bases

Abstract

We establish new conditions that prevent the existence of (weak) normal integral bases in tame Galois extensions of number fields. This leads to the following result: under appropriate technical hypotheses, the existence of a normal integral basis in the upper layer of an abelian tower Q \subset K \subset L forces the tower to be split in a very strong sense.

17 pages, 4 figures, uses xypic, minor revisions following referee's report. To appear in Annales de L'Institut Fourier