On the parity of the number of irreducible factors of self-reciprocal polynomials over finite fields

Using the Stickelberger–Swan theorem, the parity of the number of irreducible factors of a self-reciprocal even-degree polynomial over a finite field will be hereby characterized. It will be shown that in the case of binary fields such a characterization can be presented in terms of the exponents of the monomials of the self-reciprocal polynomial.