István Gaál

On the resolution of index form equations in biquadratic number fields, II

In this paper we develop a method for computing all small solutions (i.e. with coordinates of absolute value <107) of index form equations in totally real biquadratic number fields. If the index form equation is not solvable, this will also be recognized by our algorithm in most cases. As an application we present all such solutions in quadratic extensions K of Q(√5) of discriminant DKQ < 63000 and of Q(√2) of discriminant DKQ < 39000.

Computing all power integral bases in orders of totally real cyclic sextic number fields

An algorithm is given for determining all power integral bases in orders of totally real cyclic sextic number fields. The orders considered are in most cases the maximal orders of the fields. The corresponding index form equation is reduced to a relative Thue equation of degree 3 over the quadratic subfield and to some inhomogeneous Thue equations of degree 3 over the rationals. At the end of the paper, numerical examples are given.

Computing all power integral bases of cubic fields

Applying Bakers effective method and the reduction procedure of Baker and Davenport, we present several lists of solutions of index form equations in (totally real and complex) cubic algebraic number fields. These solutions yield all power integral bases of these fields. 1. Introduction. Let K be a cubic algebraic number field and denote by ZK the ring of integers of K. A power integral basis of K is an integral basis of the form {1, &, &2} with some a E ZK. If there exists such an a, then we say that K is monogenic, since ZK = Z(&e). Obviously, if a has this property, then it holds also for a + k with any k E Z. From a practical point of view, it is important to know whether there exists a power integral basis of K, and if so, what are the numerical values of a. If the Galois group of K is cyclic, then the discriminant of K is a full square. The problem of monogeneity in cyclic cubic fields was considered by M. N. Gras (12), (13), Archinard (1) and Dummit and Kisilevsky (4). M. N. Gras and Archinard gave necessary and sufficient conditions for monogeneity and tested for several numerical examples whether or not the field is monogenic. Moreover, Dummit and Kisilevsky proved that there exist infinitely many cyclic cubic fields with power integral bases. For arbitrary algebraic number fields L, it was proved by Gyory (14) that up to obvious translations by elements of Z, there are only finitely many a E ZL with ZL = Z(&), and he gave effective (but rather large) bounds for the sizes of these a. In this paper we present a method which allows us to determine all possible values of a (up to translation with rational integers) such that {1, &, &2} is a power integral

On the resolution of index form equations in sextic fields with an imaginary quadratic subfield

Abstract We give an efficient algorithm for the resolution of index form equations, especially for determining power integral bases, in sextic fields with an imaginary quadratic subfield. The method reduces the problem to the resolution of a cubic relative Thue equation over the quadratic subfield. At the end of the paper we give a table containing the generators of all power integral bases in the first 25 fields of this type with smallest discriminant (in absolute value). 1991 Mathematics Subject Classification: primary 11Y50, secondary 11Y40, 11D57

A parametric family of quintic Thue equations

For an integral parameter t ∈ Z we investigate the family of Thue equations F(x,y) = x 5 + (t - 1) 2 x 4 y - (2t 3 + 4t + 4)x 3 y 2 + (t 4 + t 3 + 2t 2 + 4t - 3)x 2 y 3 + (t 3 + t 2 + 5t + 3)xy 4 + y 5 = ±1, originating from Emma Lehmers family of quintic fields, and show that for |t| > 3.28.10 15 the only solutions are the trivial ones with x= 0 or y = 0. Our arguments contain some new ideas in comparison with the standard methods for Thue families, which gives this family a special interest.

Power integral bases in a parametric family of totally real cyclic quintics

We consider the totally real cyclic quintic fields K n = Q(v n ), generated by a root v n of the polynomial f n (x) = x 5 + n 2 x 4 - (2n 3 + 6n 2 + 10n + 10)x 3 + (n 4 + 5n 3 + 11n 2 + 15n + 5)x 2 + (n 3 + 4n 2 + 10n + 10)x + 1. Assuming that m = n 4 + 5n 3 + 15n 2 + 25n + 25 is square free, we compute explicitly an integral basis and a set of fundamental units of K n and prove that K n has a power integral basis only for n = -1, -2. For n = -1, -2 (both values presenting the same field) all generators of power integral bases are computed.

On the resolution of relative Thue equations

An efficient algorithm is given for the resolution of relative Thue equations. The essential improvement is the application of an appropriate version of Wildangers enumeration procedure based on the ellipsoid method of Fincke and Pohst.Recently relative Thue equations have gained an important application, e.g., in computing power integral bases in algebraic number fields. The presented methods can surely be used to speed up those algorithms.The method is illustrated by numerical examples.

Diophantine Equations over Global Function Fields II: R-Integral Solutions of Thue Equations

Let K be an algebraic function field over a finite field. Let L be an extension field of K of degree at least 3. Let R be a finite set of valuations of K and denote by S the set of extensions of valuations of R to L. Denote by OK,R, OL,S the ring of Rintegers of K and S-integers of L, respectively. Assume that α ∈ OL,S with L = K(α), let 0 ≠ µ ∈ OK,R, and consider the solutions (x, y) ∈ OK,R of the Thue equation NL/K (x - αy) = µ. We give an efficient method for calculating the R-integral solutions of the above equation. The method is different from that in our previous paper [Gaál and Pohst 06] and is much more efficient in many cases.

On the resolution of inhomogeneous norm form equations in two dominating variables

Applying Bakers well-known method and the reduction procedure described by Baker and Davenport, we give a numerical algorithm for finding all solutions of inhomogeneous Thue equations of type NK/Q(x + ay + X) = 1 in the variables x, y e Z and A e Zic with | A | < (max \x\, \y\))ll2, where K = Q(a) is a totally real cubic field.

Power integral bases in composits of number fields

In the present paper we consider the problem of finding power integral bases in number fields which are composits of two subfields with coprime discriminants. Especially, we consider imaginary quadratic extensions of totally real cyclic number fields of prime degree. As an example we solve the index form equation completely in a two parametric family of fields of degree 10 of this type. Received by the editors October 22, 1996. Research supported in part by Grants 16791 and 16975 from the Hungarian National Foundation for Scientific Research. AMS subject classification: Primary: 11D57: secondary: 11R21. c Canadian Mathematical Society 1998. 158