Counting monic irreducible polynomials \(P\) in \(\mathbb F_q[X]\) for which order of \(X\pmod P\) is odd (Q2642753)

\textit{H. Hasse} [Math. Ann. 166, 19--23 (1966; Zbl 0139.27501)] showed the Dirichlet density of the set of primes \(p\) for which the order of 2 modulo \(p\) is odd is 7/24. Here the density \(\delta_q\) of monic irreducible polynomials \(P\in \mathbb F_q[x]\) for which the order of \(x\) modulo \(P\) is odd is computed. \(\delta_q\) is at most 7/24 and this bound is reached exactly for \(q\) an odd power of a prime \(p\equiv 3\pmod8\). Average densities are also computed. There are two proofs pf the main result: one imitates Hasse's work and the other is elementary.