Syllabus:
Title: Arithmetic of quadratic forms
Abstract: Quadratic forms and the numbers they represent have been of interest to mathematicians for a (...)
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Title: Arithmetic of quadratic forms
Abstract: Quadratic forms and the numbers they represent have been of interest to mathematicians for a long time. For example, which integers can be expressed as a sum of two squares of integers? Or as a sum of three squares? Lagrange's four-squares theorem for instance states that any positive integer can be expressed as a sum of four squares. Such questions motivated the development of many aspects of algebraic number theory.
The goal of this seminar is to present the main results about rational quadratic forms. We will mostly follow the book of Cassels « Rational quadratic forms » and will cover the following topics: generalities about quadratic forms over fields, p-adic numbers, quadratic forms over local fields, Hasse Principle.
Bibliography: « Rational quadratic forms », J.W.S. Cassels