- 6635 South Hall
Fix a number field K with ring of integers O and a finite group G of odd order. If L/K is a Galois extension with group G, then a theorem of Erez says that the square root of the inverse different A_L/K of L/K is locally free over OG if and only if L/K is at most weakly ramified. In this case, A_L/K defines a class cl(A_L/K) in the locally free class group Cl(OG) of OG. In this talk, I will give an outline of how one can prove that the subcollection of classes realizable by tame extensions form a group when G is abelian. The main idea is to consider all Galois G-extensions instead of only field extensions and then use resolvends to characterize the realizable classes. The techniques I shall use are based on a paper of McCulloh, in which he proved the same result for the ring of integers. I will define all the terminologies that appeared in this abstract to make the talk more accessible.