In this section we extend the notion of norm to ideals. This will be
helpful in the next chapter, where
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 bound.
This is enough, because as we will see below there are only
finitely many ideals of bounded norm.

**Definition 6.3.1** (Lattice Index)
If

and

are two lattices in a vector space

, then the

*lattice index* is by definition the absolute value of the
determinant of any linear automorphism

of

such that

.

For example, if
and
, then

since multiplies
onto
.
The lattice index has the
following properties:

**Definition 6.3.2** (Norm of Fractional Ideal)
Suppose

is a fractional ideal of

. The

*norm* of

is
the lattice index

or

0 if

.

Note that if 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

by

is an automorphism of

that sends

onto

, so

**Proposition 6.3.4**
*
If and are fractional ideals, then
*

*Proof*.
By Lemma

6.3.3, it suffices to prove this when

and

are
integral ideals. If

and

are coprime, then
Theorem

5.1.4 (the Chinese Remainder Theorem) implies that

. Thus we reduce to the case when

and

for some prime ideal

and integers

.
By Proposition

5.2.3, which is
a consequence of CRT, the filtration of

given
by powers of

has successive quotients isomorphic to

.
Thus we see that

, which proves that

.

*Example 6.3.5*
We compute some ideal norms using

*SAGE*.

sage: K.<a> = NumberField(x^2 - 5)
sage: I = K.fractional_ideal(a)
sage: I.norm()
5
sage: J = K.fractional_ideal(17)
sage: J.norm()
289

We can also use functional notation:

sage: norm(I*J)
1445

We will use the following proposition in the next chapter when
we prove finiteness of class groups.

**Proposition 6.3.6**
*
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.

William Stein
2012-09-24