Universiteit Leiden

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Class invariants for tame Galois algebras

Promotores: B. Erez, P. Stevenhagen, Co-Promotor: B. de Smit

Andrea Siviero
26 juni 2013
Thesis in Leiden Repository

Let G be a finite group and K a number field with ring of integers O_K. In this thesis we study several questions related to the locally free class group Cl(O_K[G]). We mainly focus on the investigation of the set of classes in Cl(O_K[G]) which can be obtained from the ring of integers of tame G-Galois extensions of K. This set R(O_K[G]) is called the set of realizable classes. When G is abelian, R(O_K[G]) equals the Stickelberger subgroup St(O_K[G]); when G is not abelian, we just know that R(O_K[G]) is contained in St(O_K[G]), while the question if R(O_K[G]) is a group is still open. In this dissertation, after a general introduction to the subject and an exposition of the results mentioned above, we prove that St(O_K[G]) is trivial, if K is the field of rational numbers and G is a cyclic group of prime order or the dihedral group of order 2p, with p an odd prime number. Afterwards we study the functorial behavior of St(O_K[G]) under base field restriction. In the last part, restricting our attention to the abelian case, we give a result concerning the distribution of the Galois structures of the ring of integers of tame G-Galois extensions of K among realizable classes.