Classification of 3-dimensional pairs of quadratic forms over a field

Abstract

By an n-dimensional quadratic form over a field F (F-form) we mean a homogeneous quadratic polynomial in n variables with coefficients in F. We call two pairs of n-dimensional quadratic F-forms (f1,g1), (f2,g2) isomorphic if there exists a nondegenerate linear change of variables taking (f1,g1) to (f2,g2). We give a complete classification (up to isomorphism) of 3-dimensional pairs of forms (f,g) over an arbitrary field F of characteristic different from 2. We consider separately the cases of isotropic form f+tg (which is equivalent to existence of a common zero of f and g), and the anisotropic one. In the first case we make a classification depending on det(f+tg), and in the second case depending on the even Clifford algebra C0(f+tg) and det(f+tg).