JP Journal of Algebra, Number Theory and Applications
Volume 32, Issue 1, Pages 49 - 61
(February 2014)
|
|
BINOMIAL THUE EQUATIONS AND POWER INTEGRAL BASES IN PURE QUARTIC FIELDS
István Gaál and László Remete
|
Abstract: 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
We often pointed out close connection of various types of Thue equations with calculating power integral bases (cf. [9], [7]). 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 as well as we perform an extensive calculation by a high performance computer to determine “small” solutions of binomial Thue equations for
Using these results on binomial Thue equations we characterize power integral bases in infinite subfamilies of pure quartic fields. For we give all generators of power integral bases with coefficients in the integral basis.
|
Keywords and phrases: pure quartic fields, power integral bases, supercomputers. |
|
Number of Downloads: 467 | Number of Views: 1212 |
|