Let K be a number field with ring of integers OK and let G be a finite group of odd order. Given a G-Galois K-algebra Kh, let Ah be the fractional ideal in Kh whose square is the inverse different of Kh/K, which exists by Hilbert's formula since G has odd order. By a theorem of B. Erez, we know that Ah is locally free over OKG when Kh/K is weakly ramified, i.e. all of the second ramification groups in lower numbering attached to K h/K are trivial. In this case, the module Ah determines a class cl(Ah) in the locally free class group Cl(OKG) of OKG. Such a class in Cl( OKG) will be called A-realizable, and tame A-realizable if Kh/K is in fact tame. We will write A( OKG) and A t(OKG) for the sets of all A-realizable classes and tame A-realizable classes in Cl(OKG), respectively.

In this dissertation, we will consider the case when G is abelian. First of all, we will show that At( OKG) is in fact a subgroup of Cl( OKG) and that a class cl( Ah) ∈ A(O KG) is tame A-realizable if the wildly ramified primes of Kh/K satisfy suitable assumptions. Our result will imply that A(O KG) = At( OKG) holds if the primes dividing ∣ G∣ are totally split in K/Q. Then, we will show that Psi(A(O KG)) = PsiAt( OKG)) holds without any extra assumptions. Here Psi is the natural homomorphism Cl(O KG) → Cl(M(KG)) afforded by extension of scalars and Cl(M( KG)) denotes the locally free class group of the maximal OK-order M( KG) in KG. Last but not least, we will show that the group structure of At( OKG) is connected to the study of embedding problems.