Cubic Fields with Geometry

Samuel A. Hambleton, Hugh C. Williams (Autoren)

Buch | Hardcover

XIX, 493 Seiten

2018 | 1st ed. 2018
Springer International Publishing (Verlag)
978-3-030-01402-5 (ISBN)

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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.

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.

Erscheinungsdatum	21.11.2018
Reihe/Serie	CMS Books in Mathematics
Zusatzinfo	XIX, 493 p. 53 illus., 27 illus. in color.
Verlagsort	Cham
Sprache	englisch
Maße	155 x 235 mm
Gewicht	931 g
Themenwelt	Informatik ► Theorie / Studium ► Algorithmen
	Mathematik / Informatik ► Mathematik ► Analysis
	Mathematik / Informatik ► Mathematik ► Arithmetik / Zahlentheorie
	Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
Schlagworte	arithmetic matrices • binary cubic forms • Continued fractions • cubic fields • cubic Pell equation • cubic Thue equations • fundamental units • geometry of numbers • linear algebraic groups • norm equation • parametrization • simultaneous approximation • Voronoi's algorithm
ISBN-10	3-030-01402-9 / 3030014029
ISBN-13	978-3-030-01402-5 / 9783030014025
Zustand	Neuware