Characterization of primes dividing the index of a trinomial

Abstract

Let [Formula: see text] denote the ring of algebraic integers of an algebraic number field [Formula: see text], where [Formula: see text] is a root of an irreducible trinomial [Formula: see text] belonging to [Formula: see text]. In this paper, we give necessary and sufficient conditions involving only [Formula: see text] for a given prime [Formula: see text] to divide the index of the subgroup [Formula: see text] in [Formula: see text]. In particular, we deduce necessary and sufficient conditions for [Formula: see text] to be equal to [Formula: see text].

Monogenic Pisot and Anti-Pisot Polynomials

“…Theorem 2.6. [11] Let n ≥ 2 be an integer. Let K = Q(θ) be an algebraic number field with θ ∈ Z K , the ring of integers of K, having minimal polynomial f (x) = x n + Ax m + B over Q, where gcd(m, n) = d 0 , m = m 1 d 0 and n = n 1 d 0 .…”

On index divisors and monogenity of certain number fields defined by trinomials $x^6+ax+b$

“…Recently many authors have been interested on monogenity of number fields defined by trinomials. In [27,28], Khanduja et al studied the integral closedness of some number fields defined by trinomials. Their results are refined by Ibarra et al with computation of the densities (see [26]).…”

Monogenic Binomial Compositions

“…The polynomials T (x) in Theorem 1.4 are trinomials and much research has been conducted concerning the mongeneity of trinomials (see [24] and the references therein). Although necessary and sufficient conditions for a trinomial to be monogenic have been given in [22], Theorem 1.4 gives easier and more straightforward conditions to check the monogeneity of the particular trinomials in Theorem 1.4. We should also point out that there is no overlap with the trinomials in a more recent examination in [25] and the trinomials arising from Theorem 1.4.…”