The Kronecker-Vahlen theorem fails in real quadratic norm-Euclidean fields

Abstract

The present paper is devoted to methods of the automatic verifying of the Kronecker-Vahlen theorem on the shortest length of the Euclidean algorithm in algebraic number fields with the infinite group of units. We provide explicit methods to prove that the Kronecker-Vahlen theorem fails in certain algebraic number field. In particular, we give a complete solution of the problem for quadratic norm-Euclidean number fields.