Some self-dual local rings of integers not free over their associated orders

Abstract

Let p be a prime number, and let K be a finite extension of the rational p-adic field ℚp. Let L/K be a finite abelian extension with Galois group G, and let L, K denote the valuation rings of L, K respectively. Then L is a free module of rank 1 over the group algebra KG. Defining the associated order of the extension L/K byL can be viewed as a module over the ring , and a fortiori over the group ring KG.