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On the minimal index and indices of the form 2^a 3^b in a parametric family of bicyclic biquadratic felds

Let c >= 3 be integer such that c ; c - 2 ; c + 4 are square-free integers relatively prime in pairs and let Lc = Q(sqrt((c - 2) c), sqrt((c + 4) c)) be a family of bicyclic biquadratic fields. We find minimal index mu(Lc) and determine all elements with minimal index in Lc: Furthermore, we give some results concerning elements alpha with index of the form  mu(alpha) = 2^a 3^b. Precisely, we show that for every integer K >= 12 if c >= K-1 and if alpha is an element with index  mu(alpha) = 2^a 3^b <= K, then alpha is an element with minimal index  mu(alpha) = mu(Lc) = 12. We also show that for every integer C0 >= 3 we can find effectively computable integers M (C0) and N (C0) such that in case c <= C0 there are no elements alpha with index of the form  mu(alpha) = 2^a 3^b, where a > M (C0) or b > N (C0).

index form equations; minimal index; bicyclic biquadratic elds;

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