Some Results About Value Sets of Quadratic Forms Over Fields.

Abstract

In 1979, Solow defined the square class invariant of a quadratic form q over a field F to be a function from the square classes of F into the integers. For each square class of F, this function indicates the maximum number of coefficients in all diagonalized quadratic forms equivalent to q that lie in that square class. The intent of Chapter I is to determine the fields over which the square class invariant classifies quadratic forms. It will be proved that if the level of the field is at most two and if the square class invariant classifies the quadratic forms, then the field must be a C-field. Also, it will be shown that if the level of the field is at least four, then the square class invariant does not classify the quadratic forms. In 1969, Kaplansky showed that a field over which the binary quadratic form value sets have maximum index two in the multiplicative group of the field has exactly two quaternion algebras. In Chapter II a characterization will be found for all fields over which the binary form value sets have maximum index four in the multiplicative group of the field. With one exceptional case, the answer will be that the field has exactly four quaternion algebras.