Notes on geometry and arithmetic. Translated from the French by John Steinig and Constantin Manoil (Q1986981)

The goal of the present book is ``the study of arithmetical problems using geometrical methods''. The first chapter explains the rational parametrization of curves of degree \(2\) beginning with Pythagorean triples. The next four chapters develop basic algebraic geometry: affine spaces, rational points, finite fields, projective varieties (in particular conics and quadrics) and the Nullstellensatz. After an intermediate chapter on Euclidean rings, cubic surfaces and the 27 lines are discussed. The last three chapters are of a more arithmetical nature: it presents \(p\)-adic completions of the rationals, the Hasse principle, and the diophantine dimension of fields. This book is a well written introduction to classical algebraic geometry in which the necessary preliminary results from commutative algebra are provided on the fly.