Infinite families of reciprocal monogenic polynomials and their Galois groups

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 phi (2(a)q(b)), where q is an element of {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.