Z. Reichstein, B. Youssin, Conditions satisfied by characteristic polynomials in fields and division algebras , J. Pure Applied Algebra, 166 (2002), 165-184.

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.

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