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 forces the tower to be split in a very strong sense.
We present a detailed analysis of some properties of a general tamely ramified Kummer extension of number fields L/K. Our main achievement is a criterion for the existence of a normal integral basis for a general Kummer extension, which generalizes the existing results. Our approach also allows us to explicitly describe the Steinitz class of L/K and we get an easy criterion for this class to be trivial. In the second part of the paper we restrict to the particular case of tame Kummer extensions...
In this note we will determine the associated order of relative extensions of algebraic number fields, which are cyclic of prime order p, assuming that the ground field is linearly disjoint to the pth cyclotomic field, . For quadratic extensions we will furthermore characterize when the ring of integers of the extension field is free over the associated order. All our proofs are quite elementary. As an application, we will determine the Galois module structure of .
Let be a rational prime, be a finite extension of the field of -adic numbers, and let be a totally ramified cyclic extension of degree . Restrict the first ramification number of to about half of its possible values, where denotes the absolute ramification index of . Under this loose condition, we explicitly determine the -module structure of the ring of integers of , where denotes the -adic integers and denotes the Galois group Gal. In the process of determining this structure,...