Calculating power integral bases by using relative power integral bases

Abstract

Let M ⊂ K be number fields. We consider the relation of relative power integral bases of K over M with absolute power integral bases of K over Q. We show how generators of absolute power integral bases can be calculated from generators of relative ones. We apply our ideas in infinite families of octic fields with quadratic subfields.

Integral bases and monogenity of pure fields

“…We conjecture that for m = 9 + 16k the only monogenic fields are Q( 4 √ 73), Q( 4 √ 89). 3,4,5,6,8,9,10,11,12,13,14,15,16,17,19,20,21,22, 23, m = r + 25k square-free…”

“…r = 2,3,6,7,10,11,14, 15, 18, 19, 22, 23, 26, 27, 30, 31, 34, 35, 38, 39, 42, 43, 46, 47, 50, 51, 54, 55, 58, 59, 62, 63 and is not monogenic for r = 1,5,9,13, 17, 21, 25, 29, 33, 37, 41, 45, 49, 53, 57, 61, m = 5, with the exception of K = Q( 8 √ −3) which is monogenic.…”

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

“…Using the results of I.Gaál, L.Remete and T.Szabó [7] we show that the generators of relative power integral bases of O over M can be obtained from the solutions of the above relative Thue equation. As a consequence of Lemma 1 we have the following proposition which will be proved in Section 6:…”

“…Using the method of [7] we shall first determine generators of relative power integral bases of O over M and then generators of (absolute) power integral bases of O. Denote by ω (1) and ω (2) the conjugates of ω ∈ M. Let α (i) (i = 1, 2, 3) be the roots of F . Any ϑ ∈ O can be written in the form…”

“…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.…”