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Recently, the k-normal element over finite fields is defined and characterized by Huczynska et al.. In this paper, the characterization of k-normal elements, by using to give a generalization of Schwartz's theorem, which allows us to check if an element is a normal element, is obtained. In what follows, in respect of the problem of existence of a primitive 1-normal element in Fqn over Fq, for all q and n, had been stated by Huczynska et al., it is shown that, in general, this problem is not satisfied. Finally, a recursive method for constructing 1-normal polynomials of higher degree from a given 1-normal polynomial over F2m is given.

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In this paper, a computationally simple and explicit construction of some sequences of normal polynomials and self-reciprocal normal polynomials over finite fields of even characteristic are presented.

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