## LSU Historical Dissertations and Theses

1998

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

Mathematics

Jorge F. Morales

#### Abstract

In this work we compute the second Stiefel-Whitney class $w\sb2$ of the scaled trace form $Q\sb{A,a}(x)={\bf tr}\sb{A/k}(ax\sp2),$ where A is a central simple algebra over a perfect field k of characteristic different from two, $a\in A$ is a fixed element, and ${\bf tr}\sb{A/k}$ is the reduced trace. The first three chapters provide background material about quadratic forms, central simple algebras, group cohomology, and representations of linear algebraic groups. The fourth chapter presents two known results about the second Stiefel-Whitney class of trace forms: Serre's formula for the case of etale algebras and Saltman's formula for the case of central simple algebras. Our computation of $w\sb2(Q\sb{A,a})$ is done in Chapter 5. We start with the case where $A=M\sb{n}(k).$ We express $w\sb2(Q\sb{M\sb{n}(k),a})$ as a sum involving corestrictions of quaternion algebras over certain factors of $E\otimes\sb{k}E,$ where E is a commutative etale algebra over k that depends on the semisimple part of a. When A is an arbitrary central simple algebra we can find an isomorphism $\varphi:A\otimes k\sb{s}\to M\sb{n}(k\sb{s})\ (k\sb{s}$ is the separable closure of k) such that $\varphi(a):= b\in M\sb{n}(k).$ Using Galois cohomology and representation theory of reductive groups we show that$$w\sb2(Q\sb{A,a})=w\sb2(Q\sb{M\sb{n}(k),b})+{n(n-1)\over2} \lbrack A\rbrack,$$where (A) is the class of A in the Brauer group of k.

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