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

Abstract

In this paper we give some results about primitive integral elements α in the family of bicyclic biquadratic fields L_{; ; ; c}; ; ; =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(L_{; ; ; c}; ; ; )=12. We also show that for every integer C₀≥3 we can find effectively computable constants M₀(C₀) and N₀(C₀) such that if c≤C₀, than there are no elements α with index of the form μ(α)=2^{; ; ; a}; ; ; 3^{; ; ; b}; ; ; , where a>M(C₀) or b>N(C₀).