ON A CONJECTURE OF LENNY JONES ABOUT CERTAIN MONOGENIC POLYNOMIALS.

Let K = ℚ(ϑ) be an algebraic number field with ϑ satisfying a monic irreducible polynomial f (x) of degree n over ℚ. The polynomial f (x) is said to be monogenic if {1,ϑ,..., ϑ^n-1} is an integral basis of K. Deciding whether or not a monic irreducible polynomial is monogenic is an important problem in algebraic number theory. In an attempt to answer this problem for a certain family of polynomials, Jones ['A brief note on some infinite families of monogenic polynomials', Bull. Aust. Math. Soc. 100 (2019), 239-244] conjectured that if n ≥ 3, 1 ≤ m ≤ n - 1, gcd(n, mB) = 1 and A is a prime number, then the polynomial xⁿ + A(Bx + 1)^m ∈ ℤ[x] is monogenic if and only if nⁿ + (-1)^n+mBⁿ(n - m)^n-mm^m A is square-free. We prove that this conjecture is true. [ABSTRACT FROM AUTHOR]