Relative Galois module structure of integers of abelian fields

Abstract

Let L/K be an extension of algebraic number fields, where L is abelian over ℚ. In this paper we give an explicit description of the associated order 𝒜 L/K of this extension when K is a cyclotomic field, and prove that o L , the ring of integers of L, is then isomorphic to 𝒜 L/K . This generalizes previous results of Leopoldt, Chan & Lim and Bley. Furthermore we show that 𝒜 L/K is the maximal order if L/K is a cyclic and totally wildly ramified extension which is linearly disjoint to ℚ (m ' ) /K, where m ' is the conductor of K.