Infinite families of monogenic quadrinomials, quintinomials and sextinomials

Abstract

. Let f ( x ) ∈ Z [ x ] be monic, with deg( f ) = n . We say f ( x ) is monogenic if f ( x ) is irreducible over Q and { 1 , α, α 2 , . . . , α n − 1 } is a basis for the ring of integers of K = Q ( α ) , where f ( α ) = 0 . In this article, we derive a new polynomial discriminant formula, and we use it to construct infinite families of monogenic quadrinomials, quintinomials and sextinomials for any degree n ≥ 3 , 4 , 5 , respectively. These results extend previous work of the author. We also give a brief discussion concerning the adaptation of our approach beyond sextinomials.