This book introduces algebraic number theory through the problem of
generalizing 'unique prime factorization' from ordinary integers to
more general domains. Solving polynomial equations in integers
leads naturally to these domains, but unique prime factorization
may be lost in the process. To restore it, we need Dedekind's
concept of ideals. However, one still needs the supporting concepts
of algebraic number field and algebraic integer, and the supporting
theory of rings, vector spaces, and modules. It was left to Emmy
Noether to encapsulate the properties of rings that make unique
prime factorization possible, in what we now call Dedekind rings.
The book develops the theory of these concepts, following their
history, motivating each conceptual step by pointing to its
origins, and focusing on the goal of unique prime factorization
with a minimum of distraction or prerequisites. This makes a
self-contained easy-to-read book, short enough for a one-semester
course.
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