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 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 (and much
more!) is governed by the class group of , which is the quotient
of the group of fractional ideals by the principal fractional ideals
(see Chapter 7).

**Subsections**

William Stein
2012-09-24