# Norms of Ideals

In this section we extend the notion of norm to ideals. This will be helpful in proving of class groups in the next section. For example, we will prove that the group of fractional ideals modulo principal fractional ideals of a number field is finite by showing that every ideal is equivalent to an ideal with norm at most some a priori bound.

Definition 10.3.1 (Lattice Index)   If and are two lattices in vector space , then the is by definition the absolute value of the determinant of any linear automorphism of such that .

The lattice index has the following properties:

• If , then .
• If are lattices then .

Definition 10.3.2 (Norm of Fractional Ideal)   Suppose is a fractional ideal of . The of  is the lattice index

or 0 if .

Note that if is an integral ideal, then .

Lemma 10.3.3   Suppose and is an integral ideal. Then

Proof. By properties of the lattice index mentioned above we have

Here we have used that , which is because left multiplication is an automorphism of that sends onto , so .

Proposition 10.3.4   If and are fractional ideals, then

Proof. By Lemma 10.3.3, it suffices to prove this when and are integral ideals. If and are coprime, then Theorem 9.1.3 (Chinese Remainder Theorem) implies that . Thus we reduce to the case when and for some prime ideal and integers . By Proposition 9.1.8 (consequence of CRT that ), the filtration of given by powers of  has successive quotients isomorphic to , so we see that , which proves that .

Lemma 10.3.5   Fix a number field . Let be a positive integer. There are only finitely many integral ideals of with norm at most .

Proof. An integral ideal is a subgroup of of index equal to the norm of . If is any finitely generated abelian group, then there are only finitely many subgroups of of index at most , since the subgroups of index dividing an integer are all subgroups of that contain , and the group is finite. This proves the lemma.

William Stein 2004-05-06