Quadratic forms over arbitrary fields

Abstract

Introduction. Witt [5 ] proved that two binary or ternary quadratic forms, over an arbitrary field (of characteristic not 2) are equivalent if and only if they have the same determinant and Hasse invariant. His proof is brief and elegant but uses a lot of the theory of simple algebras. The purpose of this note is to make this fundamental theorem more accessible by giving a short proof using only the general results of modern theory of cohomology of finite groups. Professor Artin (Princeton lectures, 1956) gave such a proof for fields k over which local class field theory holds. For such fields, the group of values of the quadratic norm residue symbol is cyclic of order 2 while for arbitrary fields it may be any abelian group of exponent 2. So our proof is necessarily different from his; still it owes much to his methods.