We prove a new irreducibility theorem for a particular class of polynomials, and we use it to construct infinite families of reciprocal monogenic polynomials. These results extend previous work on reciprocal sextic polynomials to reciprocal polynomials of degree φ(2^aq^b), where q ∈ {3,5,7}, and a ≥ 0, b ≥ 1 are integers. As an application, we construct infinite families of reciprocal monogenic polynomials with prescribed Galois group.

The author thanks the referee for a very thorough reading of the manuscript, and for the many positive comments and suggestions.