## UE Algebraic number theory

Diplômes intégrant cet élément pédagogique :

### Descriptif

Algebraic number theory studies algebraic numbers (i.e., complex numbers that are roots of polynomials with integer coefficients) using techniques and tools from algebra.

Its origins probably go back to the work of Gauss, Eisenstein, Jacobi and its development over the centuries, under the impulse of Dirichlet, Dedkind, Hilbert and many others, is motivated by the study of some important problems in number theory, such as generalizations of the quadratic reciprocity law or the famous Fermat’s last theorem.

Algebraic number theory has now become one of the essential areas of number theory, with applications ranging from diophantine geometry, factorization of integers, primality tests, and transcendence theory, but it has also enriched on its way many other fields of mathematics. For example, the search for general formulas for roots of polynomials has been a powerful source for the development of group theory, thanks to the work of Ruffini, Abel, up to the magnificent Galois theory.

This course aims to study number fields which are extensions of finite degree of the field of rational numbers, the structure of their ring of integers and its prime ideals (ramification theory) and their completions with respect to archimedean and non-archimedean valuations (the theory of p-adic fields). We will start by recalling several tools from Galois theory (for finite and infinite extensions) and, if time allows, we will end the course with an introduction to the theory of heights.

### Pré-requis recommandés

Theory of finite groups; theory of field extensions; number fields: structure, ring of integers and prime ideals, valuation theory over a number fields; basic notions in Galois theory.

### Informations complémentaires

Langue(s) : Anglais