Winfried Scharlau

Quadratic reciprocity laws

Abstract Quadratic reciprocity laws for the rationals and rational function fields are proved. An elementary proof for Hilberts reciprocity law is given. Hilberts reciprocity law is extended to certain algebraic function fields.

Quadratische Formen in additiven Kategorien

L’acces aux archives de la revue « Memoires de la S. M. F. » (http://smf. emath.fr/Publications/Memoires/Presentation.html), implique l’accord avec les conditions generales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systematique est constitutive d’une infraction penale. Toute copie ou impression de ce fichier doit contenir la presente mention de copyright.

Quadratic Forms over Formally Real Fields

The very first theorem discovered in the algebraic theory of quadratic forms was the law of inertia of Jacobi and Sylvester. This theorem is concerned with quadratic forms over the field of real numbers. In the present chapter we will be interested in various generalizations of this result, and more generally in the connections between the theory of quadratic forms and the theory of ordered fields. Our ground field will be a formally real field, that is one in which — 1 cannot be expressed as a sum of squares. The theory of these fields was developed by Artin and Schreier in a series of now classical papers. Today it is a part of basic algebra. In the years since about 1970 it has been discovered that a substantial part of this theory can be developed in a simple and elegant manner in the framework of the theory of quadratic forms.

Quadratic Forms over Fields

The algebraic theory of quadratic forms originated in a classical paper by E. Witt [1937]. The importance of this paper consists of three essential contributions to the theory. First of all Witt introduced into the theory the geometrical language which is now commonly adopted. Secondly he constructed, in a canonical fashion, a commutative ring from the collection of all regular symmetric bilinear forms over a given field. This construction proved to be of fundamental importance because it allowed mathematicians to ask new and very fruitful questions: What is the structure of this ring and what does it tell us about the forms over the given field? Finally Witt summarized, unified, and extended the then known classification theorems. Despite the landmark importance of Witt’s paper, it was only after an incubation period of almost 30 years that a vigorous development of the algebraic theory of quadratic forms began. This new development started with the appearance of Pfister’s work in 1965 and 1966 which contains above all the first deep structure theorems about the Witt ring. The beauty and the elegance of these results led immediately to new questions, problems, and results and the theory has been flourishing ever since.

Rational Quadratic Forms

Historically the theory of quadratic forms has its origins in number-theoretic questions of the following type: Which integers can be written in the form x 2 + 2y 2 , which are sums of three squares, or more generally, which integers can be represented by an arbitrary quadratic form Σ a ij x i x j integral coefficients? This general question is exceptionally difficult and we are still quite far from a complete solution. It is natural and considerably simpler to first investigate these questions over the field of rational numbers, that is, to ask for rational instead of integral solutions to the equation Σ a ij x i x j = a. This leads to the problem of classification of quadratic forms over \( \mathbb{Q} \), which was first solved by Minkowski. His solution appears in this chapter basically unaltered, except for a few simplifications and the use of modern terminology. The Gaussian sums of Gauss and Dirich-let play a significant role in the more formal algebraic part of the theory.