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In this paper, we count the number of generators of finite fields with powers of trace zero up to some point, answering a question of Z. Reichstein. More generally, we count irreducible polynomials over finite fields with some prescribed coefficients. This is done by relating this count to the problem of counting rational points on curves over finite fields whose function fields are subfields of cyclotomic function fields.
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José Felipe Voloch, Generators of finite fields with powers of trace zero and cyclotomic function fields. Port. Math. 73 (2016), no. 1, pp. 65–70DOI 10.4171/PM/1976