  Ideles and adeles can be viewed as a generalization of Minkowski theory, in
which embedding of a number field to the Cartesian product of its completions
at the archimedean valuation is generalized to an embedding of the Cartesian
product of all its completions with some restriction. This paper introduces the
basic notions of point-set topology and builds the real numbers from the
rational numbers. Then we review concepts from local fields that will lead to
the product formula and the approximation theorem. We, then, construct adeles
and ideles. The ideles modulo $k^*$ maps surjectively to the ideal class group,
and the compactness of $C_S^0$ will give rise to an alternative proof to the
Dirichlet's S-unit theorem and the finiteness of ideal class group. The paper
assumes that the reader is familiar with Dedekind domain, principal ideal
domain, unique factorization of Dedekind domain, and field norm and traces. The
first few parts of Chapter 1 in Neukirch's Algebraic Number Theory will be
sufficient for the paper.
