Conditions satisfied by characteristic polynomials in fields and division algebras

Zinovy Reichstein and Boris Youssin

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.