Power integral bases in cubic relative extensions. (English) Zbl 1014.11080

In a series of papers the author, alone or in cooperation with other authors, investigated algorithms for computing relative power integral bases in cubic, quartic, quintic and some sextic, octic and nonic fields. The enumerative method of K. Wildanger’s thesis (TU Berlin, 1997; Zbl 0912.11061) made it possible to extend these computations from cubic and quartic fields also to higher degree fields. The author and M. Pohst determined relative power integral bases in quartic relative extensions; as a result, when the base field is quadratic, all power integral bases of octic fields were determined. In the present paper the author considers the question of determining relative power bases in relative cubic extensions. The problem reduces to solving relative Thue equations with a method of the author and Pohst, which is based on the enumeration method of Wildanger.
From the author’s introduction: “We make interesting computational experiences about Wildanger’s ellipsoid method. Surprisingly the method allows to determine relative power integral bases even for sextic base fields (in the totally real case) as illustrated by the examples. For sextic base fields the resolution of the corresponding Thue equation yields solving a unit equation of \(r=12\) unknown exponents. Note that formerly such equations were solved only with at most \(r=10\) unknowns and it was not obvious that the method works with \(r>10\). The computational experiences show that \(r=12\) is very likely the limit of the method”.


11Y40 Algebraic number theory computations
11R21 Other number fields
11R04 Algebraic numbers; rings of algebraic integers


Zbl 0912.11061




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