# Quadratic Forms

These notes cover the basics of quadratic forms, expressions in x, y of the form ax^{2} + by^{2} + 2cxy, where a, b, c come from some field. The most interesting field, and the one that ties this topic in with number theory, is the field of rational numbers, **Q**.

For example x^{2} − y^{2} and xy are quadratic forms. Although they look quite different, the fact that x^{2} − y^{2} = (x + y)(x − y) means that they can be regarded as being equivalent, because we can write X = x + y and Y = x − y, in which case x^{2} − y^{2} = XY.

A fundamental tool in studying quadratic forms is the matrix where ax^{2} + by^{2} + 2cxy can be written in the form v^{T}Av for some 2 × 2 symmetric matrix, A. A fundamental problem is to classify quadratic forms up to equivalence. This is trivial over **R** where there are only 6 equivalence classes of binary quadratic forms, with representatives 0, x^{2}, x^{2}, x^{2} + y^{2}, x^{2} − y^{2} and −x^{2}− y^{2}. But, over **Q**, it is far from trivial.

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- Introduction and Contents
- CHAP01 Quadratic Forms
- CHAP02 Witt’s Decomposition
- CHAP03 Witt’s Cancellation
- CHAP04 Quaternion Algebras
- CHAP05 Pfister Forms