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.