Glasnik Matematicki, Vol. 50, No. 1 (2015), 43-63.

ON ELEMENTS WITH INDEX OF THE FORM 2a3b IN A PARAMETRIC FAMILY OF BIQUADRATIC FIELDS

Borka Jadrijević

Department of Mathematics, Faculty of Science, University of Split, Teslina 12, 21000 Split, Croatia
e-mail: borka@pmfst.hr


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)1/2, ((c+4) c)1/2) which have index of the form μ( α) =2a3b and coprime coordinates in given integral bases. Precisely, we show that if c≥11 and α is an element with index μ( α) =2a3b≤ c+1, then α is an element with minimal index μ( α) =μ( Lc) =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) =2a3b, where a>M( C0) or b>N( C0).

2010 Mathematics Subject Classification.   11D57, 11A55, 11J86, 11J68, 11Y50.

Key words and phrases.   Index form equations, minimal index, totally real bicyclic biquadratic fields, simultaneous Pellian equations, p-adic case.


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DOI: 10.3336/gm.50.1.05


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