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**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 \(F_ m\).

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 |