Quadratic Forms

These notes cover the basics of quadratic forms, expressions in x, y of the form ax2 + by2 + 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 x2 − y2 and xy are quadratic forms. Although they look quite different, the fact that x2 − y2 = (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 x2 − y2 = XY.

A fundamental tool in studying quadratic forms is the matrix where ax2 + by2 + 2cxy can be written in the form vTAv 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, x2, x2, x2 + y2, x2 − y2 and −x2− y2. But, over Q, it is far from trivial.

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