國立虎尾科技大學 |

Cubic fields with geometry

Record Type:	Language materials, printed : Monograph/item
Title/Author:	Cubic fields with geometry/ by Samuel A. Hambleton, Hugh C. Williams.
Author:	Hambleton, Samuel A.
other author:	Williams, Hugh C.
Published:	Cham :Springer International Publishing : : 2018.,
Description:	xix, 493 p. :ill. (some col.), digital ; : 24 cm.;
Contained By:	Springer eBooks
Subject:	Geometry, Algebraic. -
Online resource:	https://doi.org/10.1007/978-3-030-01404-9
ISBN:	9783030014049

Cubic fields with geometry
Hambleton, Samuel A.

Cubic fields with geometry[electronic resource] /by Samuel A. Hambleton, Hugh C. Williams. - Cham :Springer International Publishing :2018. - xix, 493 p. :ill. (some col.), digital ;24 cm. - CMS books in mathematics, ouvrages de mathematiques de la SMC,1613-5237. - CMS books in mathematics, ouvrages de mathematiques de la SMC..

Chapter 1- Cubic fields -- Chapter 2- Cubic ideals and lattices -- Chapter 3- Binary cubic forms -- Chapter 4- Construction of all cubic fields of a fixed fundamental discriminant (Renate Scheidler) -- Chapter 5- Cubic Pell equations -- Chapter 6- The minima of forms and units by approximation -- Chapter 7- Voronoi's theory of continued fractions -- Chapter 8- Relative minima adjacent to 1 in a reduced lattice -- Chapter 9- Parametrization of norm 1 elements of K -- Tables and References -- Author Index -- Symbol Index -- General Index.

The objective of this book is to provide tools for solving problems which involve cubic number fields. Many such problems can be considered geometrically; both in terms of the geometry of numbers and geometry of the associated cubic Diophantine equations that are similar in many ways to the Pell equation. With over 50 geometric diagrams, this book includes illustrations of many of these topics. The book may be thought of as a companion reference for those students of algebraic number theory who wish to find more examples, a collection of recent research results on cubic fields, an easy-to-understand source for learning about Voronoi's unit algorithm and several classical results which are still relevant to the field, and a book which helps bridge a gap in understanding connections between algebraic geometry and number theory. The exposition includes numerous discussions on calculating with cubic fields including simple continued fractions of cubic irrational numbers, arithmetic using integer matrices, ideal class group computations, lattices over cubic fields, construction of cubic fields with a given discriminant, the search for elements of norm 1 of a cubic field with rational parametrization, and Voronoi's algorithm for finding a system of fundamental units. Throughout, the discussions are framed in terms of a binary cubic form that may be used to describe a given cubic field. This unifies the chapters of this book despite the diversity of their number theoretic topics.

ISBN: 9783030014049

Standard No.: 10.1007/978-3-030-01404-9doiSubjects--Topical Terms:

580393
Geometry, Algebraic.

LC Class. No.: QA564 / .H363 2018

Dewey Class. No.: 516.35

Cubic fields with geometry
LDR:03106nam a2200337 a 4500 001 930262
003 DE-He213
005 20190503131813.0
006 m d
007 cr nn 008maaau
008 190627s2018 gw s 0 eng d
020 $a 9783030014049 $q (electronic bk.)
020 $a 9783030014025 $q (paper)
024 7 $a 10.1007/978-3-030-01404-9 $2 doi
035 $a 978-3-030-01404-9
040 $a GP $c GP
041 0 $a eng
050 4 $a QA564 $b .H363 2018
072 7 $a PBMW $2 bicssc
072 7 $a MAT012010 $2 bisacsh
072 7 $a PBMW $2 thema
082 0 4 $a 516.35 $2 23
090 $a QA564 $b .H199 2018
100 1 $a Hambleton, Samuel A. $3 1211350
245 1 0 $a Cubic fields with geometry $h [electronic resource] / $c by Samuel A. Hambleton, Hugh C. Williams.
260 $a Cham : $c 2018. $b Springer International Publishing : $b Imprint: Springer,
300 $a xix, 493 p. : $b ill. (some col.), digital ; $c 24 cm.
490 1 $a CMS books in mathematics, ouvrages de mathematiques de la SMC, $x 1613-5237
505 0 $a Chapter 1- Cubic fields -- Chapter 2- Cubic ideals and lattices -- Chapter 3- Binary cubic forms -- Chapter 4- Construction of all cubic fields of a fixed fundamental discriminant (Renate Scheidler) -- Chapter 5- Cubic Pell equations -- Chapter 6- The minima of forms and units by approximation -- Chapter 7- Voronoi's theory of continued fractions -- Chapter 8- Relative minima adjacent to 1 in a reduced lattice -- Chapter 9- Parametrization of norm 1 elements of K -- Tables and References -- Author Index -- Symbol Index -- General Index.
520 $a The objective of this book is to provide tools for solving problems which involve cubic number fields. Many such problems can be considered geometrically; both in terms of the geometry of numbers and geometry of the associated cubic Diophantine equations that are similar in many ways to the Pell equation. With over 50 geometric diagrams, this book includes illustrations of many of these topics. The book may be thought of as a companion reference for those students of algebraic number theory who wish to find more examples, a collection of recent research results on cubic fields, an easy-to-understand source for learning about Voronoi's unit algorithm and several classical results which are still relevant to the field, and a book which helps bridge a gap in understanding connections between algebraic geometry and number theory. The exposition includes numerous discussions on calculating with cubic fields including simple continued fractions of cubic irrational numbers, arithmetic using integer matrices, ideal class group computations, lattices over cubic fields, construction of cubic fields with a given discriminant, the search for elements of norm 1 of a cubic field with rational parametrization, and Voronoi's algorithm for finding a system of fundamental units. Throughout, the discussions are framed in terms of a binary cubic form that may be used to describe a given cubic field. This unifies the chapters of this book despite the diversity of their number theoretic topics.
650 0 $a Geometry, Algebraic. $3 580393
650 0 $a Equations, Cubic. $3 1211351
650 1 4 $a Algebraic Geometry. $3 670184
650 2 4 $a Number Theory. $3 672023
650 2 4 $a Algorithms. $3 527865
700 1 $a Williams, Hugh C. $3 896671
710 2 $a SpringerLink (Online service) $3 593884
773 0 $t Springer eBooks
830 0 $a CMS books in mathematics, ouvrages de mathematiques de la SMC. $3 881286
856 4 0 $u https://doi.org/10.1007/978-3-030-01404-9
950 $a Mathematics and Statistics (Springer-11649)