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István Gaál

Publications and source records attributed to István Gaál.

At least 19 recordsLinked to original sources
arXiv2026

Monogenity of pure quintic fields: the power of sieving

We provide a simple algorithm for calculating all generators of power integral bases in pure quintic fields. This procedure involves the usual standard elements like Baker's method, LLL-reduction. The main purpose of the paper is to introduce a new idea to considerably diminish the number of small exponents to be considered after the reduction step. This new idea allows to test all remaining small exponents within a few minutes, using an appropriate sieve method, which turns out to be surprisingly fast. This idea will be applicable in many similar cases.

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arXiv2025

Calculating generators of power integral bases in sextic fields with a quadratic subfield: the general case

In some previous works we gave algorithms for determining generators of power integral basis in sextic fields with a quadratic subfield, under certain restrictions. The purpose of the present paper is to extend those methods to the general case, when the relative integral basis of the sextic field over the quadratic subfield is of general form. This raises several technical difficulties, that we consider here.

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arXiv2025

Calculating generators of power integral bases in sextic fields with a real quadratic subfield

We discuss the problem of calculating generators of power integral bases in sextic fields, especially focusing on the case of sextic fields with real quadratic subfields. Our main purpose is to describe an efficient algorithm for calculating generators of power integral bases. We show that appropriately using integer arithmetics speeds up the calculations considerably. Our experiences lead to some interesting general statements on generators of power integral bases in number fields generated by a unit.

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arXiv2025

Investigating monogenity in a family of cyclic sextic fields

L. Jones characterized among others monogenity of a family of cyclic sextic polynomials. Our purpose is to study monogenity of the family of corresponding sextic number fields. This also provides the first non-trivial application of the used method, emphasizing its efficiency.

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arXiv2025

Power integral bases in a family of octic fields

Several recent results prove the monogenity of some polynomials. In these cases the root of the polynomial generates a power integral basis in the number field generated by the root. A straightforward question is whether such a number field admits other generators of power integral bases? We have investigated this problem in some previous papers and here we extend this research to a family of octic polynomials, following a recent result of L. Jones.

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arXiv2024

Monogenity and Power Integral Bases: Recent Developments

Monogenity is a classical area of algebraic number theory that continues to be actively researched. This paper collects the results obtained over the past few years in this area. Several of the listed results were presented at a series of online conferences titled "Monogenity and Power Integral Bases". We also give a collection of the most important methods used in several of these papers. A list of open problems for further research is also given.

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arXiv2024

On the monogenity of quartic number fields defined by $x^4+ax^2+b$

For any quartic number field $K$ generated by a root $α$ of an irreducible trinomial of type $x^4+ax^2+b\in Z[x]$, we characterize when $Z[α]$ is integrally closed. Also for $p=2,3$, we explicitly give the highest power of $p$ dividing $i(K)$, the common index divisor of $K$. For a wide class of monogenic trinomials of this type we prove that up to equivalence there is only one generator of power integral bases in $K=Q(α)$. We illustrate our statements with a series of examples.

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arXiv2024

On integral bases and monogenity of pure octic number fields with non-square free parameters

In all available papers, on power integral bases of pure octic number fields $K$, generated by a root $α$ of a monic irreducible polynomial $f(x)=x^8-m\in\mathbf Z[x]$, it was assumed that $m\neq \pm 1$ is square free. In this paper, we investigate the monogenity of any pure octic number field, without the condition that $m$ is square free. We start by calculating an integral basis of $\mathbf Z_K$, the ring of integers of $K$. In particular, we characterize when $\mathbf Z_K=\mathbf Z[α]$. We give sufficient conditions on $m$, which guarantee that $K$ is not monogenic. We finish the paper by investigating the case when $m=a^u$, $u\in\{1,3,5,7\}$ and $a\neq \mp 1$ is a square free rational integer.

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arXiv2024

On the monogenity of totally complex pure octic fields

Let $0,1\ne m\in Z$ and $α=\sqrt[8]{m}$. According to the results of I. Gaál and L. El Fadil, $α$ generates a power integral basis in $K=Q(α)$, if and only if $m$ is square-free and $m\not\equiv 1\;(\bmod\; 4)$. In the present paper we consider totally complex pure octic fields, that is the case $m<0$, with $m$ satisfiying the above property. In this case $(1,α,α^2,\ldots,α^7)$ is an integral basis. Our purpose is to investigate whether $K$ admits any other generators of power integral bases, inequivalent to $α$. We present an efficient method to calculate generators of power integral bases in this type of fields with coefficients $<10^{200}$ in the above integral basis. We report on the results of our calculation for this type of fields with $0>m>-5000$, which yields 2024 fields.

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arXiv2021

Monogenity in totally real extensions of imaginary quadratic fields with an application to simplest quartic fields

We describe an efficient algorithm to calculate generators of power integral bases in composites of totally real fields with imaginary quadratic fields. We show that the calculation can be reduced to solving index form equations in the original totally real fields. We illustrate our method by investigating monogenity in the infinite parametric family of imaginary quadratic extensions of the simplest quartic fields.

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arXiv2021

Calculating "small" solutions of inhomogeneous relative Thue inequalities

Thue equations and their relative and inhomogeneous extensions are well known in the literature. There exist methods, usually tedious methods, for the complete resolution of these equations. On the other hand our experiences show that such equations usually do not have extremely large solutions. Therefore in several applications it is useful to have a fast algorithm to calculate the "small" solutions of these equations. Under "small" solutions we mean the solutions, say, with absolute values or sizes $\leq 10^{100}$. Such algorithms were formerly constructed for Thue equations, relative Thue equations. The relative and inhomogeneous Thue equations have applications in solving index form equations and certain resultant form equations. It is also known that certain "totally real" relative Thue equations can be reduced to absolute Thue equations (equations over $\Bbb Z$). As a common generalization of the above results, in our paper we develop a fast algorithm for calculating "small" solutions (say with sizes $\leq 10^{100}$) of inhomogeneous relative Thue equations, more exactly of certain inequalities that generalize those equations. We shall show that in the "totally real" case these can similarly be reduced to absolute inhomogeneous Thue inequalities. We also give an application to solving certain resultant equations in the relative case.

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arXiv2021

Monogenity in totally complex sextic fields, revisited

In addition to rather complicated general methods it is interesting and valuable to develop fast efficient methods for calculating generators of power integral bases in special types of number fields. We consider sextic fields containing a real cubic and a complex quadratic fields. We develop a very simple and very efficient method to calculate generators of power integral bases in this type of fields. Our method can be applied to infinite families of number fields, as well. We substantially improve the former methods. Our algorithm is illustrated with detailed examples, involving infinite parametric families.

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arXiv2020

Power integral bases in cubic and quartic extensions of real quadratic fields

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 we only succeeded to calculate generators of power integral bases when the ground field is an imaginary quadratic field. This is the first case when we consider monogenity in the more difficult case, in extensions of real quadratic fields. We give efficient algorithms for calculating generators of power integral bases in cubic and quartic extensions of real quadratic fields, more exactly in composites of cubic and quartic fields with real quadratic fields. In case the quartic field is totally complex, we present an especially simple algorithm. We illustrate our method with two detailed examples.

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arXiv2020

Calculating relative power integral bases in totally complex quartic extensions of totally real fields

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 of complex quartic fields. In the present paper we consider power integral bases in relative extensions of totally real fields. Totally complex quartic extensions of totally real number fields seems to be the most simple case, that we detail here. As we shall see, even in this case we have to overcome several unexpected difficulties, which we can, however solve by properly (but not trivially) adjusting standard methods. We demonstrate our general algorithm on an explicit example. We describe how the general methods for solving relative index form equations in quartic relative extensions are modified in this case. As a byproduct we show that relative Thue equations in totally complex extensions of totally real fields can only have small solutions, and we construct a special method for the enumeration of small solutions of special unit equations. These statements can be applied to other diophantine problems, as well.

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arXiv2018

Totally real Thue inequalities over imaginary quadratic fields

Let $F(x,y)$ be an irreducible binary form of degree $\geq 3$ with integer coefficients and with real roots. Let $M$ be an imaginary quadratic field, with ring of integers $Z_M$. Let $K>0$. We describe an efficient method how to reduce the resolution of the relative Thue inequalities \[ |F(x,y)|\leq K \;\; (x,y\in Z_M) \] to the resolution of absolute Thue inequalities of type \[ |F(x,y)|\leq k \;\; (x,y\in Z). \] We illustrate our method with an explicit example.

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arXiv2018

Simplest quartic and simplest sextic Thue equations over imaginary quadratic fields

The families of simplest cubic, simplest quartic and simplest sextic fields and the related Thue equations are well known. The family of simplest cubic Thue equations was already studied in the relative case, over imaginary quadratic fields. In the present paper we give a similar extension of simplest quartic and simplest sextic Thue equations over imaginary quadratic fields. We explicitly give the solutions of these infinite parametric families of Thue equations over arbitrary imaginary quadratic fields.

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