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

摘要

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 O-L of L is free over the associated order A(L/K). 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.