Conditions satisfied by characteristic polynomials in fields and   division algebras
  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. 
Zinovy Reichstein (Oregon State University) and Boris Youssin   (University of the Negev, Israel)