Use of a computer scan to prove \({\mathbb{Q}}(\sqrt{2+\sqrt{2}})\) and \({\mathbb{Q}}(\sqrt{3+\sqrt{2}})\) are Euclidean. (English) Zbl 0585.12002

It is shown that the two fields given in the title are Euclidean. The technique is the usual one of showing that a fundamental region, dissected into subboxes, is covered by lattice-centred bodies, with the innovation that the norm form is majorized by a form that needs to be evaluated at at most 6 points for each subbox. Computer calculations show that the inhomogeneous minimum is not more than 0.99 (thus proving the Euclidean property): it is conjectured that the true value is \(1/2\) in each case, and is isolated from the second minimum.
Reviewer: H.J.Godwin


11R16 Cubic and quartic extensions
12-04 Software, source code, etc. for problems pertaining to field theory
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