The number of irreducible polynomials over finite fields with vanishing trace and reciprocal trace
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We present the formula for the number of monic irreducible polynomials of degree n over the finite field F_q where the coefficients of x^n-1 and x vanish for n>3. In particular, we give a relation between rational points of algebraic curves over finite fields and the number of elements a∈F_q^n for which Trace(a)=0 and Trace(a^-1)=0. Besides, we apply the formula to give an upper bound on the number of distinct constructions of a family of sequences with good family complexity and cross-correlation measure.
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