Algebraic number theory is the study of the arithmetic properties of algebraic numbers. We would like to know, for instance, if the unique factorization of elements as products of "prime elements" holds in the rings of the form Z[x] where x is an algebraic integer (such as the Gaussian integers), or better, in the whole ring of algebraic integers of a number field. This question is important in the study of diophantine equations, the most famous example being Kummer's (and Fermat's ?) approach to Fermat's last theorem, but also in many other questions, such as the theory of integral quadratic forms, the theory of normal forms of endomorphisms with integer coefficients, the theory of complex multiplication... It turns out that the unique factorization property only holds "in the sense of ideals" in general (Kummer, Dedekind), the failure being measured by a finite abelian group called the ideal class group, and whose mysteries are still at the heart of modern number theory.
This course aims at becoming familiar with those objects, by studying their properties in general and by illustrating them with concrete examples. We will thus introduce the rings of integers of number fields, we will study their structure and we will prove the property of unique factorization for ideals. We will more particularly look into the case of quadratic integers, that is rings of the form ℤ[x] with x of degree 2, and into the case of cyclotomic integers, that is rings of the form ℤ[x] with x a root of unity.
After that, we will study the so-called "geometry of numbers". Developed by Minkowski, it allows to study lattices in finite-dimensional real vector spaces. We will use this theory to prove two important theorems in algebraic number theory: the finiteness of the class group, and Dirichlet's unit theorem. Numerous applications (related for instance to the representation of integers as sums of squares or to the resolution of diophantine equations) will illustrate the course.
At the end of the course, we will move on to more advanced material: depending on the time remaining, we may for instance introduce p-adic integers and introduce the celebrated Hasse-Minkowski Theorem, or we may develop efficient tools to study integral values of quadratic forms and introduce the first steps in class field theory.
To follow this course, it is recommended to have followed the course on Galois Theory in the second year.
Bibliography
« Primes of the form x^2+ny^2 », D. A. Cox
« A Classical Introduction to Modern Number Theory », K. Ireland and M. Rosen, Springer GTM 84
« Algebraic Number Theory », J. Neukirch
« Théorie algébrique des nombres », P. Samuel, Hermann.
« Cours d'arithmétique », J.-P. Serre
« Disquisitiones arithmeticae», C. F. Gauss.
Langue du cours : français ou anglais selon la demande
- Profesor: Izquierdo Diego