EAN: 9783030437800

UNIVERSITEXT NOTES ON GEOMETRY AND ARITHMETIC...

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.

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