A computer algorithm for finding new euclidean number fields

Lecture Notes in Computer Science

Informatique théorique et applications, tome 23, n o 1 (1989), p. 101-111. <http © AFCET, 1989, tous droits réservés. L'accès aux archives de la revue « Informatique théorique et applications » implique l'accord avec les conditions générales d'utilisation (). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques Informatique théorique et Applications/Theoretical Informaties and Applications (vol.23, n° 1, 1989, p. 101 à 111) ON COMPUTATIOIMS WITH INTEGER DIVISION by Bettina JUST (*), Friedhelm MEYER AUF DER HEIDE ( 2 )

2009

Abstract. Complete lists of number fields, of given degree n and unramified outside a given finite set S of primes, are both of intrinsic interest and useful in some applications. For degrees n ≤ 5 and S = {∞, 2, 3}, the complete lists have appeared previously; there are in total 85 such fields. Here we give the complete list for n = 6 and S = {∞, 2, 3}, finding in particular exactly 398 such fields. We use a three-pronged approach to obtain this classification: an exhaustive computer search, sextic twinning, and class field theory. Also we completely identify the 2-adic and 3-adic completions of all these degree ≤ 6 fields, this information being one of the focal points of interest and essential in applications. There is a considerable literature on the classification of number fields by means of their discriminants. At present for n = 3, 4, 5, 6, 7 there are large tables, available at [B], giving all number fields of degree n and absolute discriminant less than certain bounds. It ...

In this article we describe some elements from the chapter of Mathematics " Number Theory " , as they appear in the Codex Vindobonensis phil. Gr. 65, a Byzantine Ms kept in the National Library of Austria in Vienna. This codex contains a comprehensive program of teaching Mathematics which was addressed to an audience consisting of students probably of all the grades of what is today's primary and secondary education, but also state functionaries, merchants, craftsmen of various specialities such as silversmiths and goldsmiths, builders, etc. The mathematical branch of Number Theory and the numerical position system is integrated in codex 65 in the respective chapters of the four operations and their checks. Introduction At the beginning of this article we briefly describe Codex Vindobonensis phil. Gr. 65 fols 11r ―126r, which was written circa AD 1436 by an anonymous author, and was intended for teaching purposes. We record the syllabus units so that the reader will form an initial idea about the content of the Ms, and then we analyze in detail the problems related to the Number Theory, but also more generally to the numerical system of position in Byzantium.

I systematically constructed number systems in this article which are consistent in itself.

The Mathematical Intelligencer, 1979

2021

Cover image: Postage stamp commemorating 150th birth anniversary of Richard Dedekind, whose ideas are fundamental to much of the material in this book.

2011

In 1961 J. Stein proposed an algorithm to compute the greatest common divisor of two integers. In this paper we point out that similar algorithms exist in the ring of integers of various quadratic number fields and also in the non-commutative ring of the Hurwitz quaternions. The implementations of the algorithms are straightforward. However the procedures vary from case to case. AMS Mathematics Subject Classification (2010): Primary 11A51, Secondary 11Y16 Key words and phrases: gcd in quadratic number fields, Hurwitz quater-nion, binary gcd algorithm and its extensions.

The Mathematical Intelligencer, 1980

Mathematics of Computation, 1977

The Minkowski method of unit search is applied to particular types of parallelotopes permitting to discover algebraic integers of bounded norm in a given algebraic number field of degree n at will by solving successively 2n linear inequalities for one unknown each. Application is made to the unit search for all totally real number fields of minimal discriminant for n < 7.

2000

A simple algorithm like the Euclidean algorithm that everybody knows as a method to compute the greatest common divisor of two integers, when applied in more general situa- tions may solve some unexpected problems. One can trace the fundaments of this algorithm in applications such as the recursive computation of Pade approximants, the theory of continued fractions, the recurrence relations

2018

We present a transformation, based on the Bezout&#39;s identity, which maps the set of pairs of relatively prime numbers $(p,q)$ with fixed $p$ and $0&lt;q&lt;p$, to pairs of relatively prime numbers in the $p\times p$ square in $\mathbb R^2$, in such a way that intriguing quadratic arcs show up. We exhibit parametrizations of quadratic curves which fit such quadratic arcs and we also justify algebraically the ensuing geometry.

Mathematics of Computation, 1990

This volume has been compiled in honour of the well known mathematician Hans Zasscnhaus on the occasion of his 75th birthday. As colleagues, collaborators and friends, we dedicate this work to him in the hope that it might inspire present and future researchers, in a similar fashion to the way in which his brilliant ideas filled us with the creati>e urge. Hans Zassenhaus was born in Koblenz (West Germany) on 28 May 1912 and brought up in Hamburg. There he studied mathematics under the supervision of E. Artin, E. Heeke, and E. Sperner. and was also a student of physics and biology. He was awarded his PhD at the early age of 22 with a thesis on "Kennzeichnung linearer Gruppen als Permutationsgruppen••. In the subsequent two years, which he spent at Rostock as a teaching assistant, he wrote his famous monograph on group theory which is still among th..: standard textbooks on that subject. In 1938-back in Hamburg-he qualified for a full teaching appointment with a paper on Lie rings of prime characteristic. In 1946 he was appointed associate professor and director of the Institute for Applied \1athcmatics which he had founded at Hamburg University in the same year. Accepting the challenge of an offer to help McGill University in the building up of Canadian graduate education in mathematics he left his country in 1949 for Montreal, Canada, where he was later joined by his wife and children. As Peter Redpath Professor at McGill University he supervised the PhD studies of many Canadian students. In 1957 he became a Canadian citizen. In 1959 he moved to the USA where he taught at Notre Dame (1959-1963) and at Ohio State University in Columbus. At OSU he hdd the position of research professor until his retirement five years ago. During these years he frequently visited other universities as a guest professor. We briefly mention the academic year 1955-56 at Princeton, two years at the California Institute of Technology as a Fairchild Distinguished Scholar, a Gauss professorship at Gottingen in 1967 and the US Senior Scientist Award of the Humboldt-Stiftung. In 1956 he became a Fellow of the Royal Canadian Society and in 1969 Editor-in-Chief of the Journal of Number Theory. Among the mathematicians of our time he is one of the few still active in different areas-we have already mentioned his contributions to group theory. Most graduate students learn his famous "butterfly lemma" which nowadays forms a substantial part of the proof of the Jordan-Holder-Schreier theorem. In 1978 (jointly with R. Bulow, J. Neubiiser and H. Wondraschek) he wrote a book on crystallographic groups which seems to be better known to physicists than to the mathematical community. Orders (and their ideal theory) are also among the central objectives of his research. The constructive approach clearly dominates. In a joint paper with E. C. Dade and 0. Taussky it was proven that the (n-1)st power of a fractional ideal of an order of rank n over the integers is always invertible. The authors obtained the idea of that theorem by numerical calculations on a computer and then each of them gave a different proof. This was an early and powerful demonstration of the usefulness of mathematical experimenting by computer. Later on he developed several algorithms for the embedding of an order into its maximal order. Each algorithm improved the preceding one and the numerical results obtained by each algorithm led to further theoretical improvements-0747-7171"87.040001

1996

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