Use of a computer scan to prove $\textbf {Q}(\sqrt {2+\sqrt 2})$ and $\textbf {Q}(\sqrt {3+\sqrt 2})$ are Euclidean

Abstract:

The fields in the title are shown to be norm-Euclidean by a computer scan of the unit 4-cube representing coordinates of a field element translated by integers. The method is to subdivide this cube into sufficiently many small boxes so the norm is less than unity in each box, when referred to an appropriate "neighboring" integer.