La théorie algébrique des nombres est l’étude des propriétés arithmétiques des nombres algébriques. On s’intéresse notamment à la propriété de factorisation unique des éléments comme produits d’éléments premiers dans les anneaux de la forme ℤ[x] where x is an algebraic integer (the ring of Gaussian integers ℤ[i] for example), or better yet, in the ring of all algebraic integers of a given number field. Historically, this property has played an important role in the study of Diophantine equations, for example in Kummer's famous work on Fermat's Last Theorem. It also arises in many other seemingly unrelated questions, such as the integral theory of quadratic forms, or the theory of complex multiplication. It turns out that the unique factorization property generally only persists in the sense of ideals (Kummer, Dedekind), and that its failure can be measured by a finite abelian group, the ideal class group, whose mysteries are still at the heart of modern arithmetic.
The objective of this course will initially consist of familiarizing oneself with these objects by studying their properties in generality and illustrating them with concrete examples. We will thus introduce the rings of integers of number fields, study their structure, and prove the unique factorization property of ideals. We will focus particularly on the case of quadratic integers, that is to say rings of the form ℤ[x] with x of degree 2, and on the case of cyclotomic integers, that is to say rings of the form the form ℤ[x] with x a root of unity.
Once this framework is established, we will turn our attention to what is called the geometry of numbers. Developed by Minkowski, it allows for the study of lattices in a finite-dimensional real vector space. We will use this theory to prove two important theorems in algebraic number theory: the finiteness of the ideal class group and Dirichlet's unit theorem. Numerous applications (related for example to the representation of integers as sums of squares or to the resolution of Diophantine equations) will illustrate the course.
In the second part of the course, we will tackle the proof of the famous Hasse-Minkowski theorem affirming the validity of the local-global principle for quadratic forms with rational coefficients. This will require the introduction of p-adic numbers, as well as the theory of L-functions in order to prove and use Dirichlet's theorem on arithmetic progressions.
Course Outline:
- Chapter 1: notions of commutative algebra;
- Chapter 2: number fields and rings of integers;
- Chapter 3: p-adic numbers;
- Chapter 4: quadratic forms and the Hasse-Minkowski theorem;
- Chapter 5: L-functions and Dirichlet's theorem on arithmetic progressions.
Bibliography:
- K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, Springer GTM 84;
- J. Neukirch, Algebraic Number Theory;
- P. Samuel, Théorie algébrique des nombres, Hermann;
- J.-P. Serre, Cours d’arithmétique;
- C. F. Gauss, Disquisitiones arithmeticae.
Prerequisites: To follow this course, it is recommended to have taken the second-year Galois theory course.
Examples of EA topics:
- Integers of the form x^2+ny^2;
- p-adic L-functions;
- Galois cohomology;
- Modular forms;
- The Kronecker-Weber theorem;
- Adeles and ideles;
- Hecke characters and their L-functions;
- The Prime Number Theorem;
- Apollonian circle packings.
Language : course notes in English, lectures taught in French or English depending on the students' preferences.
