## Cubic number fields with exceptional units.(English)Zbl 0732.11054

Computational number theory, Proc. Colloq., Debrecen/Hung. 1989, 103-128 (1991).
[For the entire collection see Zbl 0722.00004.]
An exceptional unit $$\eta$$ in a number field K is such that $$\eta$$-1 is also a unit. The only cubic fields K having exceptional units are the fields $$F_ m$$, defined by $$x^ 3+(m-1)x^ 2-mx-1=0,$$ and the fields $$L_ k$$, defined by $$x^ 3-kx^ 2-(k+3)x-1=0.$$ We have $$F_{-m- 3}=F_ m$$, $$L_{-k-3}=L_ k$$, and $$F_ 2=L_{-1}$$. The author is concerned only with real fields, which excludes $$F_ 0$$ and $$F_ 1$$, and so takes $$k\geq -1$$, $$m\geq 3.$$
The fields $$L_ k$$ are cyclic and in a few cases different values of k give the same field: these are listed for $$k<10000$$. In contrast no cases where $$F_ m$$ is the same for different m have been found.
The paper concentrates on the fields $$F_ m$$, and results are given on divisors of the discriminant, integral bases, and the unit group. It has been conjectured that ($$\eta$$,$$\eta$$-1) always form a fundamental pair, and this is verified for $$m\leq 500$$. An extensive table of class groups is given, as well as some properties of quadratic, cubic, and quartic extensions of $$F_ m$$.
Reviewer: H.J.Godwin (Egham)

### MSC:

 11R16 Cubic and quartic extensions 11R27 Units and factorization 11R29 Class numbers, class groups, discriminants 11Y40 Algebraic number theory computations 12F05 Algebraic field extensions

Zbl 0722.00004