Monogenic reciprocal trinomials and their Galois groups

Abstract

Let [Formula: see text] denote the cyclic group of order [Formula: see text], and let Hol([Formula: see text]) denote the holomorph of [Formula: see text]. In this paper, for any odd integer [Formula: see text], we find necessary and sufficient conditions on an integer [Formula: see text], with [Formula: see text], such that [Formula: see text] is irreducible over [Formula: see text]. When [Formula: see text] is prime and [Formula: see text] is irreducible, we show that the Galois group over [Formula: see text] of [Formula: see text] is isomorphic to either Hol([Formula: see text]) or Hol([Formula: see text]), depending on whether there exists [Formula: see text] such that [Formula: see text]. Finally, we prove that there exist infinitely many positive integers [Formula: see text] such that [Formula: see text] is irreducible over [Formula: see text] and that [Formula: see text] is a basis for the ring of integers of [Formula: see text], where [Formula: see text].