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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
##### Citations:
Zbl 0726.11022; Zbl 0726.11023
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