Conditions satisfied by characteristic polynomials in fields and division algebras

TL;DR

This paper investigates conditions under which elements in field extensions and division algebras have characteristic polynomials with zero coefficients in specified positions, revealing many cases where such elements do not exist.

Contribution

It establishes new non-existence results for elements with prescribed polynomial coefficient conditions in fields and division algebras, including the universal division algebra.

Findings

Many characteristic polynomial coefficient conditions cannot be satisfied by elements in field extensions.

The universal division algebra of degree n lacks elements with trace 0 and norm 1.

Results extend understanding of polynomial conditions in algebraic structures.

Abstract

Suppose E/F is a field extension. We ask whether or not there exists an element of E whose characteristic polynomial has one or more zero coefficients in specified positions. We show that the answer is frequently ``no''. We also prove similar results for division algebras and show that the universal division algebra of degree n does not have an element of trace 0 and norm 1.