On elements with index of the form 2a3b in a parametric family of biquadratic fields

Abstract

Abstract. In this paper we give some results about primitive integral elements α in the family of bicyclic biquadratic fields Lc = Q( (c − 2) c, (c + 4) c) which have index of the form µ (α) = 2 a 3 b and coprime coordinates in given integral bases. Precisely, we show that if c ≥ 11 and α is an element with index µ (α) = 2 a 3 b ≤ c + 1, then α is an element with minimal index µ (α) = µ (Lc) = 12. We also show that for every integer C 0 ≥ 3 we can find effectively computable constants M 0 (C 0 ) and N 0 (C 0 ) such that if c ≤ C 0 , than there are no elements α with index of the form µ (α) = 2 a 3 b , where a > M (C 0 ) or b > N (C 0 ).

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

“…. , 32 according to (5) leads to the tuples (m, u, v, α, β, γ) = (4, 32, 0, 32, 32, 32), (32, 64, 0, 64, 64, 64) (12) valid for any c (with v = 0) and to 32 tuples of special values of (m, c, u, v, α, β, γ) (with v = 0).…”

“…We also make use of an extensive calculation by Maple and Magma involving the complete resolution of thousands of Thue equations. Recently B.Jadriević investigated infinite parametric families of totally real bicyclic biquadratic number fields [10], [11], [12], determining monogenity and elements of minimal index reducing the problem to a system of Pellian equations. Here we consider a similar type of family of number fields but we use a quite different technics, involving extensive formal and numerical calculations, as well.…”

Elements with prime and small indices in bicyclic biquadratic number fields

“…Gaál and G. Nyul [5] gave an efficient algorithm for solving the Mahler type variant of the index form equation, that is, for determining elements with index divisible only by some fixed primes in bicyclic biquadratic fields. By using this method, B. Jadrijević [13] considered the existence of elements with index 2 a 3 b in one of the parametric bicyclic biquadratic fields mentioned above.…”

The Discriminant and Conductor of Bicyclic Quartic Fields

“…Work on finding other integral bases in terms of m and n has largely focused on power integral bases (see, for example, [41], [43] and [35]). Other notable work has been done to study the indices of elements of a bicyclic quartic field [29], the construction of a non-Euclidean ideal in a real bicyclic quartic field [21], sums of three squares of integral elements of a bicyclic quartic field [31] and the construction of a fundamental system of units assuming the abc conjecture [34].…”