• stably free modules
• inverse different
• Galois module structure
• Frohlich Hom-descritpion
• classgroup of locally free modules

Let G be a finite group and define the classgroup Cl(Z[G]) of the locally free modules over Z[G] (here Z is the ring of integers) as the group of the classes of locally free Z[G]-modules with the operation of the group given by the direct sum of modules and with an equivalence relation, i.e. M is equivalent to N if there exists P and Q, two finitely generated free modules, such that M+P=N+Q (here + represents the direct sum).
In this thesis we will consider the class group in the particular case when G is the Galois group of a Galois extension of number fields N/E. The purpose of this work is to study the Z[G]-modules which appears in this context, as elements of Cl(Z[G]).
In the case when N/E is a tame extension it is known that O_N (the ring of integers of N) is locally free and it class has been deeply investigated. By Taylor's Theorem(1981), this class can be expressed in terms of the Artin root number. In this thesis we consider, when it exists, the square root A(N/E) of the inverse of the different. In the case of odd degree extensions, A(N/E)
always exists and is locally free over Z[G] if and only if N/E is weakly ramified. This context has been studied by Erez(1991), who proved that if N/E is a tame, odd degree Galois extension of number fields with Galois group G, then A(N/E) is free over Z[G].
The most important instrument involved in this work is a theorem of Frohlich, known as the Hom-Description, which gives an isomorphism between the classgroup and a quotient of the group of the homomorphisms from additive group of virtual characters of G to the idele group of a number field.
Since Taylor's Theorem implies the triviality of (O_N) when N/E is tame and has odd degree, we get (A(N/E))=(O_N) and both classes are in fact trivial (we denote with (M) the class of a locally free module M in Cl(Z[G])).
It is natural to consider the problem to determine the class of A(N/E) also when the extension is tame and the degree is even. Assuming the existence of A(N/E) (which is not guaranteed in general) we can prove that, if N/E is tame, then A(N/E) is locally free over Z[G].
This case has been studied by Caputo and Vinatier(2016), where some generalizations of the result of Erez are presented. The fundamental result which we present in this thesis, is the following: the class of A(N/E) is equal to the class of O_N in the classgroup if N/E is a tame locally abelian Galois extension of number fields.
The principal tools for the proof are the Hom-description, the Stickelberger theorem, Hasse-Davenport theorem and the study of quaternionic characters. Using the previous theorem it can be proved that the class of A(N/E) is trivial, if N/E is a tame locally abelian Galois extension such that no archimedean places ramifies in N. This results follows from the study of the Artin root number of sympletic characters, combined with Taylor's Theorem.
However, the class of A(N/E) is not always trivial, even for tame locally abelian extensions.
In fact, Caputo and Vinatier, exhibit an example of tame locally abelian Galois extension of number fields whose inverse different is a square such that the class of A(N/E) is not trivial.