On prime divisors of the index of an algebraic integer

Monogenic polynomials with non-squarefree discriminant

“…We have arranged the expression in (18) so that the summands are multiplicative with respect to d. This enables us to complete the sum over all gcd(d, αβ) = 1 and factor it into an Euler product, up to some manageable error. By (18) and bounding the tail of the completed sum trivially,…”

“…Main results. Recently, Jakhar, Khanduja and Sangwan [17,18] have given necessary and sufficient conditions, based solely on n, m, A, B, for an irreducible trinomial f (x) = x n + Ax m + B to be monogenic. While examples are provided in [18] for the situation where n = tm + u, with 0 ≤ u ≤ m − 1, and either u = 0 or u is a divisor of m, no indication is given as to whether there exist infinite families of such trinomials.…”

“…Recently, Jakhar, Khanduja and Sangwan [17,18] have given necessary and sufficient conditions, based solely on n, m, A, B, for an irreducible trinomial f (x) = x n + Ax m + B to be monogenic. While examples are provided in [18] for the situation where n = tm + u, with 0 ≤ u ≤ m − 1, and either u = 0 or u is a divisor of m, no indication is given as to whether there exist infinite families of such trinomials. In this article we use the conditions given in [18], together with some asymptotic results (see Theorem 8 and Theorem 9), to construct infinite families of monogenic trinomials f (x) = x n + Ax m + B, where n ≥ 2 is an integer and m ≥ 1 is a proper divisor of n. We point out that the elements of the families given here are of a different form than previously examined in the literature.…”

“…[18] Let n ≥ 2 be an integer. Let K = Q(θ) be an algebraic number field with θ ∈OK , the ring of integers of K, having minimal polynomial f (x) = x n + Ax m + B over Q, where m ≥ 1 is a proper divisor of n. Let t = n/m.…”

Monogenic fields arising from trinomials

“…As this work was in final edits for release, the authors were made aware of an overlapping recent parallel research line. Using an extension of Dedekind's index criterion, Jakhar, Khanduja, and Sangwan ( [10] and [11]) established necessary and sufficient conditions for any trinomial to be monogenic. Their work is more complete than our Theorems 3.5 and 3.6, but our methods are distinct (we use the Montes Algorithm).…”