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.
The case of "quadratic integers", that is of Z[x] where x is an integer, is historically the most important one, and will be discussed in details. We will see that its arithmetic is related to the problem of classifiying binary integral quadratic forms (Lagrange, Gauss, Dedekind) and to the elementary looking problem of deciding which integers are represented by a given form. For instance, we know since Fermat that if p is a prime with p = 1 mod 4, then p is a sum of two square, or that if p = 1,3,7,9 mod 20, then p is (exclusively) either of the form x2 + 5 y2 or of the form 2 x2 + 2 xy + 3 y2 (Euler, Lagrange). Along the way, we will provide efficient tools to prove this kind of results, including the notion of "genus" of a quadratic form (Lagrange, Gauss), which is the starting point of the famous class field theory.
Bibliography
« A Classical Introduction to Modern Number Theory », K. Ireland and M. Rosen, Springer GTM 84
« Théorie algébrique des nombres », P. Samuel, Hermann.
« Disquisitiones arithmeticae», C. F. Gauss.
Course language : Français
ECTS credits: 5
- Profesor: Izquierdo Diego