Splitting Patterns and Invariants of Quadratic Forms

Abstract

AbstractLet φ be an anisotropic quadratic form over a field F of characteristic not 2. The splitting pattern of φ is defined to be the increasing sequence of nonnegative integers obtained by considering the Witt indices iW(φk) of φ over K where K ranges over all field extensions of F. Restating earlier results by HURRELBRINK and REHMANN, we show how the index of the Clifford algebra of φ influences the splitting pattern. In the case where F is formally real, we investigate how the signatures of φ influence the splitting behaviour. This enables us to construct certain splitting patterns which have been known to exist, but now over much “simpler” fields like formally real global fields or ℝ(t). We also give a full classification of splitting patterns of forms of dimension less than or equal to 9 in terms of properties of the determinant and Clifford invariant. Partial results for splitting patterns in dimensions 10 and 11 are also provided. Finally, we consider two anisotropic forms φ and φ of the same dimension m with φ ⟂ − φ ∈ In F and give some bounds on m depending on n which assure that they have the same splitting pattern.