Dedekind Domains and Unique Factorization of Ideals

Unique factorization into irreducible elements frequently fails for rings of integers of number fields. In this chapter we will deduce a central property of the ring of integers $\O_K$ of an algebraic number field, namely that every nonzero ideal factors uniquely as a products of prime ideals. Along the way, we will introduce fractional ideals and prove that they form a free abelian group under multiplication. Factorization of elements of $\O_K$ (and much more!) is governed by the class group of $\O_K$, which is the quotient of the group of fractional ideals by the principal fractional ideals (see Chapter 7).