Inequalities Between Pythagoras Numbers and Algebraic Ranks in Witt Rings of Fields.

1999

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Mathematics

Robert Perlis

Abstract

This dissertation establishes new lower bounds for the algebraic ranks of certain Witt classes of quadratic forms. Let K denote a field of characteristic different from 2 and let q be a quadratic form over K. The form q is said to be algebraic when q is Witt equivalent to the trace form qL∣K of some finite algebraic field extension L∣K . When q is algebraic, the algebraic rank of q is defined to be the degree of the minimal extension L∣K whose trace form is Witt equivalent to q. It is an important, unsolved problem to find reasonable bounds on the algebraic rank of a given algebraic form. This dissertation makes a beginning contribution to this problem of bounding the algebraic rank by investigating a simple, special case. Let sigma denote a totally positive square-class in a field K (characteristic different from 2, as always). Assuming that the 1-dimensional quadratic form sigma X2 is algebraic, what can be said about the algebraic rank? Even in this simple case, little was known prior to this dissertation. Let pyKs denote the pythagoras number of sigma relative to K. This is the smallest natural number, j, such that or can be written as a sum of j squares of elements in K. Let n be the algebraic rank of sigma X2. In general, n is unknown, and a reasonable lower bound is sought. The main result is: pyKs ≤2n-dn+1 where d(n + 1) is the sum of the coefficients of the 2-adic expansion of n + 1. Thus, if sigma is chosen with a large pythagoras number, then the algebraic rank n must be correspondingly large as well. This dissertation also considers a slightly more general case.

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