LAUSR

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 ( f 1 , g 1 ) , ( f 2 , g 2 ) isomorphic if there exists a nondegenerate linear change of variables taking ( f 1 , g 1 ) to ( f 2 , g 2 ) . 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 + t g (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 + t g ) , and in the second case depending on the even Clifford algebra C 0 ( f + t g ) and det ( f + t g ) .