A computer algorithm for finding new Euclidean number fields
The main result of this article is the construction of more than 1200 number fields of degrees \(4\), \(5\) and \(6\) that are Euclidean with respect to the absolute value of the norm. The main idea is classical: divide the fundamental domain of the lattice \({\mathcal O}_K\) into sufficiently small cubes and, for each cube, find an element \(\gamma\in O_K\) such that \(| N(\xi-\gamma)|<1\) for all \(\xi\) in that cube. If there are cubes left after several repeated divisions, stage 2 of the algorithm checks whether multiplication by units transforms a cube into some covered region: these cubes can then be eliminated. The author goes on to discuss applications of his algorithm to rings of \(S\)-integers and finally asks whether the inhomogeneous minima of number fields with fixed signature can be bounded by a constant times the square root of the absolute value of the discriminant; as examples in degree \(3\) and \(4\) due to Swinnerton-Dyer and Davenport show (see the reviewer's survey [\textit{F. Lemmermeyer}, Expo. Math. 13, No. 5, 385-416 (1995; Zbl 0843.11046)]), the answer is negative.
scientific article; zbMATH DE number 1229646