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In their recent article Chang et al. [Chang, Y., Troung, T. K., Reed, I. S. (2001). Normal bases over GF(q). J Algebra 241:89-101] have determined all those extensions of Galois fields for which the normal basis generators are characterized by the (obviously necessary) property of having nonzero trace. In the Present article, we present a simpler proof of a generalization of that result and discuss an application concerning the existence of trace-compatible sequences of primitive normal bases for certain primary closures of Galois fields.