In this note an improvement of Katz's bound on the number of elements in a finite field with given trace and norm is given. The improvement is obtained by reducing the problem to estimating the number of rational points on certain toric Calabi-Yau hypersurfaces, and then to use detailed cohomological calculations by Rojas-Leon and the second author for such toric hypersurfaces. © 2010 Walter de Gruyter Berlin · New York.