# Thursday 08 Feb 2018: Number Theory Seminar: Galois actions on units of rings of integers

### Alex Torzewski - University of Warkick

**H103** 14:30-16:30

Given a finite Galois extension K/Q, the units of the ring of integers of K canonically define a representation of the Galois group of K/Q. If we extend scalars to Q, then its isomorphism class is determined by the signatures of the intermediate subfields of K/Q. It is much less clear what arithmetic properties are carried by the isomorphism class of M itself. We shall show that for some families of number fields, the isomorphism class of M is determined by data involving only class groups and ramification information. This relies on studying the abstract properties of regulator constants. These are invariants of representations and occur in practice in a variety of settings.