Monogenic reciprocal trinomials and their Galois groups

摘要

Let C-n denote the cyclic group of order n, and let Hol(C-n) denote the holomorph of C-n. In this paper, for any odd integer m >= 3, we find necessary and sufficient conditions on an integer A, with |A| = 3, such that F-m,F-A(x) = x(2m) + A(xm) + 1 is irreducible over Q. When m = q >= 3 is prime and F-q,F-A(x) is irreducible, we show that the Galois group over Q of F-q,F-A(x) is isomorphic to either Hol(C-q) or Hol(C-2q), depending on whether there exists y is an element of Z such that A(2) - 4 = qy(2). Finally, we prove that there exist infinitely many positive integers A such that F-q,F-A(x) is irreducible over Q and that {1, theta, theta(2),..., theta(2q-1)} is a basis for the ring of integers of K = Q(theta), where F-q,F-A(theta) = 0.