Related papers: Calculating power integral bases by solving relati…
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 $K=M(\alpha)$, where $\alpha$ is a root of $f$. This is an infinite…
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…
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…
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…
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…
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…
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…
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…
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…
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…
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.
We obtain a polynomial type upper bound for the size of the integral solutions of Thue equations $F(X,Y) = b$ defined over a totally real number field $K$, assuming that $F(X,1)$ has a root $\alpha$ such that $K(\alpha)$ is a CM-field.…
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 $c\in\mathbb{Z}_{M},$ $c\notin\left\{ 0,\pm2\right\}$. We…
We exactly determine the integral solutions to a previously untreated infinite family of cubic Thue equations of the form $F(x,y)=1$ with at least $5$ such solutions. Our approach combines elementary arguments, with lower bounds for linear…
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)…
Let $K$ be an imaginary quadratic field other than $\mathbb{Q}(\sqrt{-1})$ and $\mathbb{Q}(\sqrt{-3})$. We construct relative power integral bases between certain abelian extensions of $K$ in terms of Weierstrass units.
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…
We will give upper bounds for the number of integral solutions to quartic Thue equations. Our main tool here is a logarithmic curve $\phi(x, y)$ that allows us to use the theory of linear forms in logarithms. This manuscript improves the…
An elementary, albeit higher dimensional, argument is used to compute the area under the power function curve between 0 and 1.
We describe several algorithms for computing $e$-th roots of elements in a number field $K$, where $e$ is an odd prime-power integer. In particular we generalize Couveignes' and Thom\'e's algorithms originally designed to compute…