Corpus ID: 119580247

# Relative power integral bases in infinite families of quartic extensions of quadratic field

@article{Gaal2013RelativePI,
title={Relative power integral bases in infinite families of quartic extensions of quadratic field},
author={Istv'an Ga'al and T'i mea Szab'o},
journal={arXiv: Number Theory},
year={2013}
}
• Published 27 September 2018
• Mathematics
• arXiv: Number Theory
We consider infinite parametric families of octic fields, that are quartic extensions of quadratic fields. We describe all relative power integral bases of the octic fields over the quadratic subfields.
13 Citations
Quartic Relative Extensions
• I. Gaál
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• Diophantine Equations and Power Integral Bases
• 2019
In this chapter we consider quartic relative extensions, relative power integral bases in these extensions, and also the absolute power integral bases in the extension field by using the relativeExpand
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• 2016
Let $M\subset K$ 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 absoluteExpand
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• Acta Scientiarum Mathematicarum
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Investigations of monogenity and power integral bases were recently extended from the absolute case (over Q) to the relative case (over algebraic number fields). Formerly, in the relative case weExpand
Power integral bases in a family of sextic fields with quadratic subfields
• Mathematics
• 2015
Abstract Let M = Q(i √d) be any imaginary quadratic field with a positive square-free d. Consider the polynomial f(x) = x3 − ax2 − (a + 3)x − 1 with a parameter a ∈ ℤ. Let K = M(α), where α is a rootExpand
CALCULATING RELATIVE POWER INTEGRAL BASES IN TOTALLY COMPLEX QUARTIC EXTENSIONS OF TOTALLY REAL FIELDS
• I. Gaál
• Mathematics
• JP Journal of Algebra, Number Theory and Applications
• 2019
Some time ago we extended our monogenity investigations and calculations of generators of power integral bases to the relative case. Up to now we considered (usually totally real) extensions ofExpand
Binomial Thue equations and power integral bases in pure quartic fields
• Mathematics
• 2018
It is a classical problem in algebraic number theory to decide if a number field admits power integral bases and further to calculate all generators of power integral bases. This problem isExpand
Calculating Power Integral bases by Solving Relative Thue Equations
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• 2014
Abstract In our recent paper I. Gaál: Calculating “small” solutions of relative Thue equations, J. Experiment. Math. (to appear) we gave an efficient algorithm to calculate “small” solutions ofExpand
Non-Monogenity of an Infinite Family of Pure Octic Fields
Let m be a square free integer. The aim of this paper is to prove that the infinite family of pure octic field L Q √m is non-monogenic if m mod , ultimately, to complete the classification of pureExpand
Computing relative power integral bases in a family of quartic extensions of imaginary quadratic fields
• Mathematics, Physics
• 2015
Let $M$ be an imaginary quadratic field with the ring of integers $\mathbb{Z}_{M}$ and let $\xi$ be a root of polynomial $$f\left( x\right) =x^{4}-2cx^{3}+2x^{2}+2cx+1,$$ where $c\in\mathbb{Z}_{M},$Expand
The Scheme of Monogenic Generators and its Twists
Given an extension of algebras B/A, when is B generated by a single element θ ∈ B over A? We show there is a scheme MB/A parameterizing the choice of a generator θ ∈ B, a “moduli space” ofExpand

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