Relative Galois Module Structure of Rings of Integers of Absolutely Abelian Number Fields

We define an extension L/K of absolutely abelian number fields to be Leopoldt if the ring of integers OL of L is free as a module over the associated order AL/K of L/K. Furthermore, we say that an abelian number field K is Leopoldt if every extension L/K with L/Q abelian is Leopoldt. In this paper, we make progress towards a classification of Leopoldt number fields and extensions. The two main results are as follows. Let Q denote the nth cyclotomic field and Q its maximal real subfield. Let p be an odd prime and n ∈ N. Then (i) any sub-extension of Q(pn)/Q of degree a power of p is Leopoldt and (ii) Q n)+ is Leopoldt. In addition, we give a partial classification of Leopoldt subfields of Q n) and note that this classification is in fact complete when p− 1 is square-free.