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On the resolution of index form equations in biquadratic number fields. III: The bicyclic biquadratic case. (English) Zbl 0853.11026
The authors continue their research on the practical resolution of index form equations in biquadratic number fields [cf. Zbl 0726.11022, Zbl 0726.11023]. Here, the authors consider totally real bicyclic biquadratic number fields, i.e. number fields \(K= \mathbb{Q}( \sqrt {m}, \sqrt {n})\) where \(m\), \(n\) are distinct, square-free positive integers; such fields have Galois group \(V_4\) (the Klein four group). The index of an algebraic integer \(\alpha\in K\) is the index of \(\mathbb{Z} [\alpha ]\) in the ring of integers in \(K\). The authors determine the minimal index \(\mu (K)\) that can occur, and all integers of \(K\) having that minimal index, for all totally real, bicyclic biquadratic fields of discriminant \(D< 10^6\). Fixing an integral basis \(1, \omega_2, \omega_3, \omega_4\) of \(K\), one can express the index of \(\alpha= x_1+ x_2 \omega_2+ x_3 \omega_3+ x_4 \omega_4\) with \(x_1, \dots, x_4\in \mathbb{Z}\) as \(|I(x_2, x_3, x_4) |\), where \(I\) is a homogeneous polynomial in \(\mathbb{Z}[ x_2, x_3, x_4]\) of degree 6 (the index form); hence determining the integers in \(K\) of given index \(a\) is equivalent to solving \(I(x_2, x_3, x_4)= \pm a\). The authors show that this can be reduced to solving simultaneous Pellian equations, by using that in the particular case considered here, after a suitable linear transformation on \(x_2, x_3, x_4\), \(I\) can be factored into three binary quadratic forms of positive discriminant with coefficients in \(\mathbb{Z}\). Then they solve the system of Pellian equations, using lower bounds for linear forms in logarithms and a modification of the reduction lemma of Baker and Davenport.

MSC:
11D99 Diophantine equations
11Y40 Algebraic number theory computations
11R16 Cubic and quartic extensions
11R80 Totally real fields
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