In commutative algebra, the norm of an ideal is a generalization of a norm of an element in the field extension. It is particularly important in number theory since it measures the size of an ideal of a complicated number ring in terms of an ideal in a less complicated ring. When the less complicated number ring is taken to be the ring of integers, Z, then the norm of a nonzero ideal I of a number ring R is simply the size of the finite quotient ring R/I.
Let A be a Dedekind domain with field of fractions K and integral closure of B in a finite separable extension L of K. (this implies that B is also a Dedekind domain.) Let
I
A
and
I
B
be the ideal groups of A and B, respectively (i.e., the sets of nonzero fractional ideals.) Following the technique developed by Jean-Pierre Serre, the norm map
N
B
/
A
:
I
B
→
I
A
is the unique group homomorphism that satisfies
N
B
/
A
(
q
)
=
p
[
B
/
q
:
A
/
p
]
for all nonzero prime ideals
q
of B, where
p
=
q
∩
A
is the prime ideal of A lying below
q
.
Alternatively, for any
b
∈
I
B
one can equivalently define
N
B
/
A
(
b
)
to be the fractional ideal of A generated by the set
{
N
L
/
K
(
x
)
|
x
∈
b
}
of field norms of elements of B.
For
a
∈
I
A
, one has
N
B
/
A
(
a
B
)
=
a
n
, where
n
=
[
L
:
K
]
. The ideal norm of a principal ideal is thus compatible with the field norm of an element:
N
B
/
A
(
x
B
)
=
N
L
/
K
(
x
)
A
.
Let
L
/
K
be a Galois extension of number fields with rings of integers
O
K
⊂
O
L
. Then the preceding applies with
A
=
O
K
,
B
=
O
L
, and for any
b
∈
I
O
L
we have
N
O
L
/
O
K
(
b
)
=
O
K
∩
∏
σ
∈
Gal
(
L
/
K
)
σ
(
b
)
,
which is an element of
I
O
K
. The notation
N
O
L
/
O
K
is sometimes shortened to
N
L
/
K
, an abuse of notation that is compatible with also writing
N
L
/
K
for the field norm, as noted above.
In the case
K
=
Q
, it is reasonable to use positive rational numbers as the range for
N
O
L
/
Z
since
Z
has trivial ideal class group and unit group
{
±
1
}
, thus each nonzero fractional ideal of
Z
is generated by a uniquely determined positive rational number. Under this convention the relative norm from
L
down to
K
=
Q
coincides with the absolute norm defined below.
Let
L
be a number field with ring of integers
O
L
, and
a
a nonzero (integral) ideal of
O
L
. The absolute norm of
a
is
N
(
a
)
:=
[
O
L
:
a
]
=
|
O
L
/
a
|
.
By convention, the norm of the zero ideal is taken to be zero.
If
a
=
(
a
)
is a principal ideal, then
N
(
a
)
=
|
N
L
/
Q
(
a
)
|
.
The norm is completely multiplicative: if
a
and
b
are ideals of
O
L
, then
N
(
a
⋅
b
)
=
N
(
a
)
N
(
b
)
. Thus the absolute norm extends uniquely to a group homomorphism
N
:
I
O
L
→
Q
>
0
×
,
defined for all nonzero fractional ideals of
O
L
.
The norm of an ideal
a
can be used to give an upper bound on the field norm of the smallest nonzero element it contains: there always exists a nonzero
a
∈
a
for which
|
N
L
/
Q
(
a
)
|
≤
(
2
π
)
s
|
Δ
L
|
N
(
a
)
,
where
Δ
L
is the discriminant of
L
and
s
is the number of pairs of (non-real) complex embeddings of L into
C
(the number of complex places of L).