arXiv subjects
Publications and source records attributed to István Gaál.
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.
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 is especially delicate to consider in an infinite parametric family of number fields. In the present paper we investigate power integral bases in the infinite parametric family of pure quartic fields $K=Q(\sqrt[4]{m})$. We often pointed out close connection of various types of Thue equations with calculating power integral bases. In this paper we determine power integral bases in pure quartic fields by reducing the corresponding index form equation to quartic binomial Thue equations. We apply recent results on the solutions of binomial Thue equations of type $a^4-mb^4=\pm 1\; ({\rm in}\; a,b\in Z)$ as well as we perform an extensive calculation by a high performance computer to determine "small" solutions (with $\max(|a|,|b|)<10^{500}$) of binomial Thue equations for $m<10^7$. Using these results on binomial Thue equations we characterize power integral bases in infinite subfamilies of pure quartic fields. For $1< m< 10^7$ we give all generators of power integral bases with coefficients $<10^{1000}$ in the integral basis.
It is a classical problem in algebraic number theory to decide if a number field is monogeneous, that is if it admits power integral bases. It is especially interesting to consider this question in an infinite parametric familiy of number fields. In this paper we consider the infinite parametric family of simplest quartic fields $K$ generated by a root $ξ$ of the polynomial $P_t(x)=x^4-tx^3-6x^2+tx+1$, assuming that $t>0$, $t\neq 3$ and $t^2+16$ has no odd square factors. In addition to generators of power integral bases we also calculate the minimal index and all elements of minimal index in all fields in this family.
Diophantine equations can often be reduced to various types of classical Thue equations. These equations usually have only very small solutions, on the other hand to compute all solutions (i.e. to prove the non-existence of large solutions) is a time consuming procedure. Therefore it is very practical to have a fast algorithm to calculate the "small" solutions, especially if "small" means less than e.g. $10^{100}$. Such an algorithm was constructed by A.Peth\H o in 1987 based on continued fractions. In the present paper we construct a similar type of fast algorithm to calculate "small" solutions of {\it relative} Thue equations. Our method is based on the LLL reduction algorithm. We illustrate the method with explicit examples. The algorithm has several applications.
Let $m$ be a square-free positive integer, $m\equiv 2,3 \; (\bmod \; 4)$. We show that the number field $K=Q(i,\sqrt[4]{m})$ is non-monogene, that is it does not admit any power integral bases of type $\{1,α,\ldots,α^7\}$. 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.
Let $M=Q(i\sqrt{d})$ be any imaginary quadratic field with a positive square-free $d$. Consider the polynomial \[ f(x)=x^3-ax^2-(a+3)x-1, \] with a parameter $a\in Z$. Let $K=M(α)$, where $α$ is a root of $f$. This is an infinite parametric family of sextic fields depending on two parameters, $a$ and $d$. Applying relative Thue equations we determine the relative power integral bases of these sextic fields over their quadratic subfields. Using these results we also determine generators of (absolute) power integral bases of the sextic fields.
We consider binomial Thue equations of type $x^n-my^n=\pm 1$ in $x,y\in Z$. Optimizing the method of Peth\H o we perform an extensive calculation by a high performance computer to determine all solutions with $\max(|x|,|y|)<10^{500}$ of binomial Thue equations for $m<10^7$ for exponents $n=3,4,5,7,11,13,17,19,23,29$.
In our recent paper we gave an efficient algorithm to calculate "small" solutions of relative Thue equations (where "small" means an upper bound of type $10^{500}$ for the sizes of solutions). Here we apply this algorithm to calculating power integral bases in sextic fields with an imaginary quadratic subfield and to calculating relative power integral bases in pure quartic extensions of imaginary quadratic fields. In both cases the crucial point of the calculation is the resolution of a relative Thue equation. We produce numerical data that were not known before.
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 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.
We consider infinite parametric families of high degree number fields composed of quadratic fields with pure cubic, pure quartic, pure sextic fields and with the so called simplest cubic, simplest quartic fields. We explicitly describe an integral basis of the composite fields. We construct the index form, describe their factors and prove that the monogenity of the composite fields imply certain divisibility conditions on the parameters involved. These conditions usually can not hold, which implies the non-monogenity of the fields. The fields that we consider are higher degree number fields, of degrees 6 up to 12. The non-monogenity of the number fields is stated very often as a consequence of the non-existence of the solutions of the index form equation. Up to our knowlegde it is not at all feasible to solve the index form equation in these high degree fields, especially not in a parametric form. On the other hand our method implies directly the non-monogenity in almost all cases. We obtain our results in a parametric form, characterizing these infinite parametric families of composite fields.
Let $m$ be an integer, $m\neq -8,-3,0,5$ such that $m^2+3m+9$ is square free. Let $α$ be a root of \[ f=x^6-2mx^5-(5m+15)x^4-20x^3+5mx^2+(2m+6)x+1. \] The totally real cyclic fields $K=Q(α)$ are called simplest sextic fields and are well known in the literature. Using a completely new approach we explicitly give an integral basis of $K$ in a parametric form and we show that the structure of this integral basis is periodic in $m$ with period length 36. We prove that $K$ is not monogenic except for a few values of $m$ in which cases we give all generators of power integral bases.
Let $m$ be a square-free integer ($m\neq 0,\pm 1$). We show that the structure of the integral bases of the fields $K=Q(\sqrt[n]{m})$ are periodic in $m$. For $3\leq n\leq 9$ we show that the period length is $n^2$. We explicitly describe the integral bases, and for $n=3,4,5,6,8$ we explicitly calculate the index forms of $K$. This enables us in many cases to characterize the monogenity of these fields. Using the explicit form of the index forms yields a new technic that enables us to derive new results on monogenity and to get several former results as easy consequences. For $n=4,6,8$ we give an almost complete characterization of the monogenity of pure fields.