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.
In the first part of the course, we will get familiarized with all those objects, by studying their general properties and by illustrating them through concrete examples. We will introduce the rings of integers of number fields, we will study their structure and we will prove the unique factorization of ideals. We will particularly focus on the case of quadratic integers, ie rings of the form ℤ[x] with x of degree 2, and on the case of cyclotomic integers, ie rings of the form ℤ[x] with x a root of unity. Once we have settled this framework, we will be interested in the so-called "geometry of numbers". Developed by Minkowski, it allows to study lattices in finite dimensional real vector spaces. We will then use that theory to prove an important theorem of algebraic number theory: the finiteness of the ideal class group. Many applications (related for instance to the representation of integers as sums of squares or to the resolution of Diophantine equations) will illustrate the course.
In the second part, we will focus on objects that encode local information in number theory, the so-called p-adic fields. We will define them, study their basic algebraic and analytic properties, and prove two fundamental results: Hensel's lemma and the weak approximation theorem.
Finalle, in the third part of the course, we will develop tools of analytic number theory. We will study the Riemann zeta function and the Dirichlet L-functions. That will allow us to prove Dirichlet's theorem on arithmetic progressions.
We will conclude the course by proving the Hasse-Minkowski theorem, which is a beautiful application of all the tools we will have developed in the previous parts of the course.
To follow this course, it is recommended to have followed the course on Galois theory in second year.
Examples of EA topics
-Cébotarev's theorem.
-Diophantine dimension of fields.
-Integers of the form x^2+ny^2.
-S-unit equation.
-Galois cohomology.
-Arithmetic dynamics.
-Kronecker-Weber theorem.
Bibliography
« 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.
Language of the course : French or English, depending on the demand
