This book deals with several aspects of what is now called “explicit number
theory,” not including the essential algorithmic aspects, which are for the
most part covered by two other books of the author [Coh0] and [Coh1]. The
central (although not unique) theme is the solution of Diophantine equa-
tions, i.e., equations or systems of polynomial equations that must be solved
in integers, rational numbers, or more generally in algebraic numbers. This
theme is in particular the central motivation for the modern theory of arith-
metic algebraic geometry. We will consider it through three of its most basic
aspects.