RATIONAL JORDAN DECOMPOSITION OF BILINEAR FORMS

Abstract

This is a continuation of our previous work on Jordan decomposition of bilinear forms over algebraically closed fields of characteristic 0. In this note, we study Jordan decomposition of bilinear forms over any field K0 of characteristic 0. Let V0 be an n-dimensional vector space over K0. Denote by [Formula: see text] the space of bilinear forms f : V0 × V0 → K0. We say that f = g + h, where f, g, [Formula: see text], is a rational Jordan decomposition of f if, after extending the field K0 to an algebraic closure K, we obtain a Jordan decomposition over K. By using the Galois group of K/K0, we prove the existence of rational Jordan decompositions and describe a method for constructing all such decompositions. Several illustrative examples of rational Jordan decompositions of bilinear forms are included. We also show how to classify the unimodular congruence classes of bilinear forms over an algebraically closed field of characteristic different from 2 and over the real field.