Infinite families of monogenic trinomials and their Galois groups

Abstract

Let [Formula: see text] with [Formula: see text]. Let [Formula: see text] and [Formula: see text] denote, respectively, the symmetric group and alternating group on [Formula: see text] letters. Let [Formula: see text] be an indeterminate, and define [Formula: see text] where [Formula: see text] are certain prescribed forms in [Formula: see text]. For a certain set of these forms, we show unconditionally that there exist infinitely many primes [Formula: see text] such that [Formula: see text] is irreducible over [Formula: see text], [Formula: see text], and the fields [Formula: see text] are distinct and monogenic, where [Formula: see text]. Using a different set of forms, we establish a similar result for all square-free values of [Formula: see text], with [Formula: see text], and any positive integer value of [Formula: see text] for which [Formula: see text] is square-free. Additionally, in this case, we prove that [Formula: see text]. Finally, we show that these results can be extended under the assumption of the [Formula: see text]-conjecture. Our methods make use of recent results of Helfgott and Pasten.