On power basis of a class of algebraic number fields

Abstract

Let [Formula: see text] be an algebraic number field with [Formula: see text] in the ring [Formula: see text] of algebraic integers of [Formula: see text] and [Formula: see text] be the minimal polynomial of [Formula: see text] over the field [Formula: see text] of rational numbers. In 1977, Uchida proved that [Formula: see text] if and only if [Formula: see text] does not belong to [Formula: see text] for any maximal ideal [Formula: see text] of the polynomial ring [Formula: see text] (see [Osaka J. Math. 14 (1977) 155–157]). In this paper, we apply the above result to obtain some necessary and sufficient conditions involving the coefficients of [Formula: see text] for [Formula: see text] to equal [Formula: see text] when [Formula: see text] is a trinomial of the type [Formula: see text]. In the particular case when [Formula: see text], it is deduced that [Formula: see text] is an integral basis of [Formula: see text] if and only if either (i) [Formula: see text] and [Formula: see text] or (ii) [Formula: see text] divides [Formula: see text] and [Formula: see text].

Monogenic polynomials with non-squarefree discriminant

“…In consideration of U (X) = U M (X) + U E (X) and lines (17) and (19) with the choice X 0 = X 1/4 , the proof of Theorem 8 is complete.…”

“…For more information on general monogenic fields, see [23]. For some recent specific examinations of monogenic fields, see [1,2,4,6,8,11,12,13,19,30]. We see from (1) that K being monogenic is equivalent to the existence of some irreducible polynomial f (x), with f (θ) = 0 and K = Q(θ), such that ∆(f ) = ∆(K).…”

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]).…”

“…According to Jones's definition, a monic irreducible polynomial F(x) ∈ Z[x] is monogenic if Z[α] is integrally closed, where α is a complex root of F(x). Remark that the results given in [27,28,29,30,31,32], can only decide on the integral closedness of Z[α], but cannot test whether the field is monogenic or not. The monogenity of polynomials can be used partially in the study of monogenity of number fields, but the converse is not true; because it is possible that a number field generated by a complex root α of a non monogenic polynomial can be monogenic.…”

Monogenic Binomial Compositions

“…For any fixed degree n ≥ 2, the density of monogenic 1074 Joshua Harrington and Lenny Jones polynomials is 6/π 2 ≈ .607927 [4]. However, determining infinite families of degree-n monogenic polynomials can be difficult, and much research has been done to locate such families [1,2,5,8,10,12,13,16,17,23,29].…”