Notes on Geometry and Arithmetic

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Sprache: Englisch

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Beschreibung

This English translation of Daniel Coray¿s original French textbook Notes de géométrie et d¿arithmétique introduces students to Diophantine geometry. It engages the reader with concrete and interesting problems using the language of classical geometry, setting aside all but the most essential ideas from algebraic geometry and commutative algebra. Readers are invited to discover rational points on varieties through an appealing ¿hands on¿ approach that offers a pathway toward active research in arithmetic geometry. Along the way, the reader encounters the state of the art on solving certain classes of polynomial equations with beautiful geometric realizations, and travels a unique ascent towards variations on the Hasse Principle.

Highlighting the importance of Diophantus of Alexandria as a precursor to the study of arithmetic over the rational numbers, this textbook introduces basic notions with an emphasis on Hilbert¿s Nullstellensatz over an arbitrary field. A digression on Euclidian rings is followed by a thorough study of the arithmetic theory of cubic surfaces. Subsequent chapters are devoted to p-adic fields, the Hasse principle, and the subtle notion of Diophantine dimension of fields. All chapters contain exercises, with hints or complete solutions.

Notes on Geometry and Arithmetic will appeal to a wide readership, ranging from graduate students through to researchers. Assuming only a basic background in abstract algebra and number theory, the text uses Diophantine questions to motivate readers seeking an accessible pathway into arithmetic geometry.

Über den Autor

Daniel François Coray (1947-2015) was Professor of Mathematics at the University of Geneva. His research interests spanned algebraic number theory, arithmetic of algebraic varieties, enumerative geometry, theoretical applications of computer methods and mathematical modeling of spatial concepts. His achievements in research were matched by his passion for teaching throughout his career. Daniel Coray passed away unexpectedly in June 2015 in Geneva only two months after he had completed the French version of this book manuscript.

Zusammenfassung

Offers readers a 'hands on' introduction to Diophantine geometry

Progresses from introductory to advanced topics using the language of classical geometry

Assumes only modest prerequisites in abstract algebra and number theory

Contains numerous exercises throughout

Inhaltsverzeichnis

Chapter 1. Diophantus of Alexandria.- Chapter 2. Algebraic closure; affine space.- Chapter 3. Rational points; finite fields.- Chapter 4. Projective varieties; conics and quadrics.- Chapter 5. The Nullstellensatz.- Chapter 6. Euclidean rings.- Chapter 7. Cubic surfaces.- Chapter 8. p-adic completions.- Chapter 9. The Hasse principle.- Chapter 10. Diophantine dimension of fields.

Details

Erscheinungsjahr:	2020
Fachbereich:	Arithmetik & Algebra
Genre:	Mathematik , Medizin , Naturwissenschaften , Technik
Rubrik:	Naturwissenschaften & Technik
Medium:	Taschenbuch
Inhalt:	xii 181 S. 118 s/w Illustr. 3 farbige Illustr. 181 p. 121 illus. 3 illus. in color.
ISBN-13:	9783030437800
ISBN-10:	3030437809
Sprache:	Englisch
Einband:	Kartoniert / Broschiert
Autor:	Coray, Daniel
Übersetzung:	Steinig, John Manoil, Constantin
Auflage:	1st edition 2020
Hersteller:	Springer International Publishing Springer International Publishing AG
Verantwortliche Person für die EU:	Springer Verlag GmbH, Tiergartenstr. 17, D-69121 Heidelberg, juergen.hartmann@springer.com
Maße:	235 x 155 x 11 mm
Von/Mit:	Daniel Coray
Erscheinungsdatum:	07.07.2020
Gewicht:	0,348 kg

Artikel-ID: 118023402