|M.Sc Student||Jonathan Charbit|
|Subject||On p-Groups and K-Admissibility|
|Department||Department of Mathematics||Supervisor||Full Professor Sonn Jack|
Let K be a field, L a finite extension of K and G a finite group. L is said to be K-adequate if there is a division algebra D finite dimensional and central over K containing L as a maximal subfield. We say that G is K-admissible if there is a Galois extension L/K with Galois group G such that L is K-adequate.
In the case of number fields, K-admissibility could be characterized by an arithmetic criterion: G is K-admissible if and only if there exists a Galois extension L/K with Galois group G such that for every prime p that divides the order of G, there are two primes of K such that their local Galois group contains a p-Sylow subgroup of G.
In this work, we investigate the K-admissibility of some p-groups over number fields. We present basic facts on K-admissibility and we explain why we always have to assume that K contains at least two prime divisors of p. We present theorems of Grunwald-Wang-Hasse and Neukirch regarding Grunwald-Neukirch problems, we describe the structure of the Galois group of the maximal p-extension of a p-adic field, we state a few theorems of Saltman on generic extension theory and we give a basic introduction to embedding problems theory, in particular Brauer embedding problems where a local-global principle holds. Connecting all those facts together, we are led to general results on abelian p-groups and semi-direct product of p-groups.
We also prove, using a local-global principle for Brauer embedding problems, that in the case of groups of order 16 over quadratic number fields, the main conjecture for K-admissibility holds.
Relying on the structure theorem for extra-special p-groups, we show that the main conjecture for K-admissibility also holds for extra-special p-groups over any number field.
Finally, we introduce the concept of local realizability that allows us to unify a large part of our results and to state new results.