Exceptional units and Euclidean number fields
Let \(K\) be an algebraic number field and \({\mathcal O}^{*}\) be it's group of units. An element \(\varepsilon \in {\mathcal O}^{*}\) is an exceptional unit if and only if \(1-\varepsilon \in {\mathcal O}^{*}\). H. W. Lenstra has shown that if there exist sufficiently many exceptional units with the property that all of their differences are also units, then the number field \(K\) is Euclidean with respect to the norm. Using this result, the author gives in this paper two methods for computing a subset of \({\mathcal O}^{*}\) such that the difference of two elements of this subset is also in \({\mathcal O}^{*}\). In particular, one method is based on graph theory algorithm for the maximum clique problem. His computations yields \(42\) new Euclidean number fields with degrees 8, 9, 10, 11 and 12.
Euclidean number fields, Lenstra's constant
scientific article; zbMATH DE number 5162041