The number of irreducible polynomials over finite fields with vanishing trace and reciprocal trace

We present the formula for the number of monic irreducible polynomials of degree n over the finite field $${\mathbb {F}}_q$$ where the coefficients of $$x^{n-1}$$ and x vanish for $$n\ge 3$$ . In particular, we give a relation between rational points of algebraic curves over finite fields and the number of elements $$a\in {\mathbb {F}}_{q^n}$$ for which Trace $$(a)=0$$ and Trace $$(a^{-1})=0$$ .