Notes on Geometry and Arithmetic. 1st ed. 2020

種類:

電子ブック

責任表示:

by Daniel Coray

出版情報:

Cham : Springer International Publishing : Imprint: Springer, 2020

著者名:

• Coray, Daniel.
• SpringerLink (Online service)

シリーズ名:

Universitext ;

ISBN:

9783030437817 [3030437817]    

注記:

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.
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 t

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