Date of Award

1995

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics

First Advisor

Jack Ohm

Abstract

This work consists of results on three questions in the algebraic theory of forms. The first question deals with characterizing the Witt kernel (i.e. the anisotropic non-singular quadratic forms over that become hyperbolic) over a given field extension. For separable quadratic and bi-quadratic extension this is well known (for example see (B1, 4.2 and 4.3), (B2, p. 121), (L, p. 200), (ELW, 2.12)). In chapter 2, we provide answers to this question for inseparable quadratic and bi-quadratic extensions. We provide theorem 2.1.5, which in particular answers question 4.4 in (B2). From this result we prove the excellence property for inseparable quadratic extension, which is in turn used to find the Witt kernels of inseparable bi-quadratic extensions. In the third chapter we study the relation between similarity of quadratic forms and isomorphism and place equivalence of their function fields. In sections 3.1 and 3.2, we show that the function fields of special Pfister neighbors of the same Pfister form are isomorphic. Also we show that any Pfister neighbor of codimension $\le$4 is special; in particular this implies place equivalence implies birational equivalence in this case. Together with the main result of (H3), this gives an affirmative answer of the quadratic Zariski problem in dimension 3. (see S 3.3). In S 3.4 we provide few results on the problem of descent of similarity over field extensions and some examples where similarity is determined by their generic splitting tower. In the last chapter we provide a positive answer for the following conjecture of Pfister-Leep in the special case d = the characteristic of the field k C scONJECTURE. For a fixed d, if k is a $C\sbsp{0}{d}$-field, then k is a p-field for some prime $p \ne d.$.

Pages

63

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