Monogenic Binomial Compositions

Abstract

We say a monic polynomial f (x) ∈ Z[x] of degree n ≥ 2 is monogenic if f (x) is irreducible over Q and {1, θ, θ 2 , . . . , θ n−1 } is a basis for the ring of integers of Q(θ), where f (θ) = 0. In this article, we investigate when a pair of polynomials f (x) = x n −a and g(x) = x m −b has the property that f (x) and f (g(x)) are monogenic.

A note on non-monogenity of number fields arised from sextic trinomials

“…As an application, they provided infinitely many monogenic trinomials. Further in 2020, Jones and Harrington [6] investigated few pairs of binomials f (x) = x n − a and g(x) = x m − b having the property that both f (x) and f (g(x)) are monogenic. In fact, they provided necessary and sufficient conditions for the monogenicity of the polynomial (x m − b) n − a with n = 2, 3.…”

Monogenity and Power Integral Bases: Recent Developments

“…• J. Harrington and L. Jones [84] gave conditions for the monogenity of (x m − b) n − a the composition of x n − a and x m − b • A. Jakhar, R. Kalwaniya and P. Yadav [96] consider monogenity of (x m − b) n − a, the composition of x n − a and x m − b using a refined version of the Dedekind criterion • J. Harrington and L. Jones [85] considers monogenity of Φ p a (Φ 2 b (x)), where Φ N (x) is the cyclotomic polynomial of index N • L. Jones [112] considers monotonically stable polynomial of type g( f n (x))…”

Monogenity and Power Integral Bases: Recent Developments

“…J. Harrington and L. Jones [109] gave conditions for the monogenity of (x m − b) n − a, and the composition of x n − a and x m − b; • Jakhar, A. R. Kalwaniya and P. Yadav [110] considered monogenity of (x m − b) n − a, and the composition of x n − a and x m − b using a refined version of the Dedekind criterion; • J. Harrington and L. Jones [111] considered monogenity of Φ p a (Φ 2 b (x)), where Φ N (x) is the cyclotomic polynomial of index N; • L. Jones [112] considered monotonically stable polynomials of type g( f n (x)); • L. Jones [113] constructed infinite collections of monic Eisenstein polynomials f (x), such that f (x d n ) are monogenic for all integers n ≥ 0 and d > 1; • L.Jones [114] considered monogenity of S k (x p ), where S k (x) = x 3 − kx 2 − (k + 3)x − 1 the Shanks polynomial; • L. Jones [115] considered monogenity of f (x p ), where f (x) is the characteristic polynomial of an Nth order linear recurrence; • J. Harrington and L. Jones [116] gave conditions for the monogenity of f (x p n ), where f (x) = x m + ax m−1 + b; • S. Kaur, S. Kumar and L. Remete [117] considered monogenity of f (x k ), where f (x) =…”