Rational Quadratic Forms

Description

The material of the book is largely nineteenth century but the treatment is structured by two twentieth century insights. The first, which seems to have come to its full recognition in the work of Hasse and Witt, is that the theory of forms over fields is logically simpler and more complete than that over rings. It is therefore appropriate, contrary to what seemed natural earlier, to study forms with rational coefficients and under rational equivalence before attacking integral forms and integral equivalence. The second major insight, due to Hensel and Hasse, is the perspective introduced by the p-adic view-point. This unveils the majestic simplicity of the logical structure: in particular it banishes for ever the need for the plethora of multifarious “characters” and “invariants” which earlier (and some later) authors use to distinguish forms which are p-adically inequivalent. The p-adic numbers are as natural as the reals—indeed it can plausibly be argued that they are logically simpler and that it is only by indoctrination that we feel that the reals are more familiar. However, as the p-adic numbers are not yet as well- known as they should be to the broad audience to which this book is addressed, no knowledge of them has been presupposed.