ON ELEMENTS WITH INDEX DIVISIBLE BY FIXED PRIMES IN A PARAMETRIC FAMILY OF BICYCLIC BIQUADRATIC FIELDS
Borka Jadrijević · 2014
In this talk we will present 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 μ(α) divisible by fixed primes 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, then there are no elements α with index of the form μ(α)=2^{; ; a}; ; 3^{; ; b}; ; , where a>M(C₀) or b>N(C₀).