Chapter 1
A Brief Classical
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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