Determining Elements of Minimal Index in an Infinite Family of Totally Real Bicyclic Biquadratic Number Fields

Abstract

Let c = 2 be a positive integer such that c and c + 4 are squarefree. We consider the infinite parametric family of bicyclic biquadratic * The first author is supported in part by K115479 from the Hungarian National Foundation for Scientific Research † Both authors were supported in part by the Croatian Science Foundation under the project no. 6422. 2010 Mathematics Subject Classification: Primary 11R04; Secondary 11Y50 Key words and phrases: bicyclic biquadratic fields, power integral basis, minimal index fields K = Q(√ 2c, 2(c + 4)). We determine the integral basis of the field. We show that K admits no power integral basis, determine the minimal index and all elements of minimal index. We use the solutions of a parametric family of quartic Thue equations and extensive numerical calculations by Maple and Magma are also involved.

Two families of monogenic $S_4$ quartic number fields

“…When c and c+ 4 are square-free, Gaál and Jadrijević [10] show Q √ 2c, 2(c + 4) is not monogenic, compute an integral basis, and determine the elements of minimal index.…”

Elements with prime and small indices in bicyclic biquadratic number fields

“…B. Jadrijević [11], [12], partly in common with I. Gaál [4], determined the field index, minimal index and elements with minimal index in four parametric families of totally real bicyclic biquadratic fields.…”