On the resolution of index form equations in dihedral quartic number fields. (English) Zbl 0823.11074

Gaál, István; Pethö, Attila; Pohst, Michael

Exp. Math. 3, No. 3, 245-254 (1994).

Let \(K\) be a totally real quartic number field whose Galois group is the dihedral group \(D_ 8\). \(K\) can be obtained in the form \(K= L(\sqrt {\mu})\), where \(L\) is a real quadratic field and \(\mu\) is an algebraic integer of \(L\) which is not a square in \(L\). Let \(\{1, \omega\}\) be an integral basis of \(L\) and suppose that \(K/L\) has a relative integral basis \(\{1, \psi\}\) (this is the case e.g. if \(L\) has class number 1); then \(\{1, \omega, \psi, \omega\psi\}\) is an integral basis of \(K\). If \(\alpha= x_ 1+ x_ 2 \omega+ x_ 3 \psi+ x_ 4 \omega \psi\) with \(x_ 1, \ldots, x_ 4\in \mathbb{Z}\) then the index of \(\mathbb{Z} [\alpha]\) in the ring of integers of \(K\) can be expressed as \(I(x_ 2, x_ 3, x_ 4)\), where \(I\) is some homogeneous polynomial of degree 6 with rational integral coefficients, called the index form with respect to the basis \(\{1, \omega, \psi, \omega \psi\}\). \(I\) is independent of \(x_ 1\) since for every \(x_ 1\in \mathbb{Z}\) we have \(\mathbb{Z} [\alpha- x_ 1]= \mathbb{Z}[ \alpha]\).
The authors consider for given \(J\) the so-called index form equation \[ I(x_ 2, x_ 3, x_ 4)= J \quad \text{in} \quad x_ 2, x_ 3, x_ 4\in \mathbb{Z}. \tag{1} \] The authors attack equation (1) by reducing it to an equation of the type \[ G_ n= x^ 2+D \quad \text{in} \quad n,x\in \mathbb{Z}, \tag{2} \] where \(G_ n\) is a second-order linear recurrence sequence and \(D\) is a given integer. Subsequently, the authors try to solve (2) by means of some elementary sieving method which consists of sieving out certain residue classes for \(n\) and then determining the solutions in the remaining classes by computing a number of Jacobi symbols. The authors mention that their method does not always work but that with their sieving procedure they could solve equation (1) for about 70% of the totally real quartic fields of Galois group \(D_ 8\) having discriminants less than \(10^ 6\).
In several other papers, the authors considered index form equations for other types of quartic fields, and attacked these by a much more involved and much more time-consuming method, using among other things lower bounds for linear forms in logarithms, [cf. J. Number Theory 38, 18-34 (1991; Zbl 0726.11022) and 35-51 (1991; Zbl 0726.11023)].

Reviewer: J.-H.Evertse (Leiden)

MSC:

11Y40	Algebraic number theory computations
11D72	Diophantine equations in many variables
11R16	Cubic and quartic extensions
11B37	Recurrences

Keywords:

quartic fields; index form equation; second-order linear recurrence sequence

Citations:

Zbl 0726.11022; Zbl 0726.11023

References:

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