Chapter 1

A Brief Classical

Introduction

A development in graph theory or topology from last July may already

be viewed as "ancient history," but in the theory of quadratic forms, and

indeed throughout number theory, "ancient" really means ANCIENT. For

example, Babylonian tablets dating back to 1900-1600 BC suggest that it

was then known that there are infinitely many primitive Pythagorean

triples: solutions (a, 6, c) £ 1? of the equation x\ + x\ — x\ — 0, with

gcd(a, 6, c) — 1; and it was known how to produce them. These results could

be said to constitute the first theorem in quadratic forms; and historians

suggest that this is the first instance on record in which mathematics was

clearly being done for fun, not for commercial purposes or to determine

property boundaries.

Historical Remark. Pythagoras actually lived during the period 580-500

BC, long after the above discovery to which we apply his name. See Boyer

[Bo] for a thorough treatment of this early history.

We begin with the classical definition of quadratic forms. Later on we

will see that it can be advantageous to view a quadratic form as a mapping

on a module bearing an inner product.

1.1. Quadratic Forms as Polynomials

Definition 1.1. Let R be an integral domain of characteristic not 2, and

suppose F is a field containing R. An n-ary quadratic form over F is a

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http://dx.doi.org/10.1090/gsm/090/01