Non-monogenity in a family of octic fields

Abstract

Let m be a square-free positive integer, m ≡ 2, 3 (mod 4). We show that the number field K = Q(i, 4 √ m) is non-monogene, that is it does not admit any power integral bases of type {1, α, . . . , α 7 }. In this infinite parametric family of Galois octic fields we construct an integral basis and show non-monogenity using only congruence considerations.Our method yields a new approach to consider monogenity or to prove non-monogenity in algebraic number fields. It is well applicable in parametric families of number fields. We calculate the index of elements as polynomials depending on the parameter, factor these polynomials and consider systems of congruences according to the factors.

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

“…If µ = 2 and τ ≥ 4, then N + φ 1 (F) = S 11 + S 12 has two sides such that S 12 is of degree 1 and S 11 is of degree 2 and its attached residual polynomial of F(x) is R (15,14), (24,23), (42, 41), (51, 50), (69, 68), (78, 77)} (mod 81), then N + φ 2 (F) = S 21 + S 22 has two sides joining (0, v), (1, 2), (2, 1), and (3, 0) with v ≥ 4, R 1 (F)(y) = y 2 − y ± 1 the residual polynomial of F(x) associated to S 22 and S 21 is of degree 1. Since R 1 (F)(y) is irreducible over F 3 , there are exactly three prime ideals of Z K lying above 3.…”

“…There methods are based on the arithmetic of the index form equations. They studied monogenity of several algebraic number fields (see [2,17,14,15,16,34]). Recently many authors have been interested on monogenity of number fields defined by trinomials.…”

The Monogeneity of Kummer Extensions and Radical Extensions

“…Chang [2] completely describes the monogeneity of the Kummer extension K when [K : Q] = 6. Gaál and Remete [9] investigate [K : Q] = 8. Though we do not outline it further here, there is a wealth of literature on monogenic abelian extensions of a fixed degree.…”

Power integral bases in a family of sextic fields with quadratic subfields

“…Recently I.Gaál, L.Remete and T.Szabó [7] studied the relation of monogenity and relative monogenity which was already applied in a family of octic fields by I.Gaál and L.Remete [6]. In the present paper we utilize similar tools to describe power integral bases in a well known infinite parametric family of sextic fields.…”