The Kronecker-Vahlen theorem fails in real quadratic norm-Euclidean fields. (May 2021)

• Record Type: Journal Article
• Authors: Vaskouski, Maksim; Kondratyonok, Nikita
• Is Part Of: Journal of symbolic computation. Volume 104(2021)
• Page Start: 134
• Page End: 141
• Publication Date: 2021-05
• Subjects: Division chain -- Euclidean minimum -- Norm-Euclidean quadratic number field; Mathematics -- Data processing -- Periodicals; Numerical analysis -- Data processing -- Periodicals; Automatic programming (Computer science) -- Periodicals; Mathématiques -- Informatique -- Périodiques; Analyse numérique -- Informatique -- Périodiques; Programmation automatique -- Périodiques; Automatic programming (Computer science); Mathematics -- Data processing; Numerical analysis -- Data processing; Periodicals; Electronic journals; 510.285
• DOI: 10.1016/j.jsc.2020.04.009
• Languages: English
• ISSNs: 0747-7171

Abstract

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.