Relative Galois module structure of rings of integers of absolutely abelian number fields

Abstract

Let L/K be an extension of number fields where L/ℚ is abelian. We define such an extension to be Leopoldt if the ring of integers L of L is free over the associated order . Furthermore we define an abelian number field K to be Leopoldt if every finite extension L/K with L/ℚ abelian is Leopoldt in the sense above. Previous results of Leopoldt, Chan & Lim, Bley, and Byott & Lettl culminate in the proof that the n-th cyclotomic field ℚ(n) is Leopoldt for every n. In this paper, we generalize this result by giving more examples of Leopoldt extensions and fields, along with explicit generators.