# Normal Integral Bases for kummer extensions of number fields

**Luogo:** Dipartimento di Matematica, via Sommarive, 14 - Povo (TN) - Aula Seminari

**Ore** 10:00

**Relatore:**

**Ilaria Del Corso****(**Università di Pisa**)**

**Abstract**

The aim of this talk is to introduce the audience to the Galois module theory, a branch of algebraic number theory which studies the rings of integers of Galois extensions of number fields as modules over the integral group ring of the Galois group. The classical theory has as starting point the Normal Basis Theorem for a Galois extension L/K, and there is a natural question asking if, in the case of number fields, an analogous result hods for the ring of integers: namely, whether or not there exists a normal integral basis (NIB). In the first part of my talk, I shall introduce the classical approach, the fundamental results, and I will give a brief account of the modern developments of this theory. In the second part, I will present some results (part of some joint papers with Lorenzo P. Rossi) on the existence of the NIB for a general tamely ramified Kummer extension of number fields L/K. Our approach allows to obtain also an explicit description of the Steinitz class of L/K and will give an easy test for the existence of a NIB in some particular cases.

**Referente: **Massimiliano Sala