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 and 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**

is the unique group homomorphism that satisfies

for all nonzero prime ideals of *B*, where is the prime ideal of *A* lying below .

Alternatively, for any one can equivalently define to be the fractional ideal of *A* generated by the set of field norms of elements of *B*.^{[1]}

For , one has , where .

The ideal norm of a principal ideal is thus compatible with the field norm of an element:

^{[2]}

Let be a Galois extension of number fields with rings of integers .

Then the preceding applies with , and for any we have

which is an element of .

The notation is sometimes shortened to , an abuse of notation that is compatible with also writing for the field norm, as noted above.

In the case , it is reasonable to use positive rational numbers as the range for since has trivial ideal class group and unit group , thus each nonzero fractional ideal of is generated by a uniquely determined positive rational number.
Under this convention the relative norm from down to coincides with the absolute norm defined below.

Let be a number field with ring of integers , and a nonzero (integral) ideal of .

The absolute norm of is

By convention, the norm of the zero ideal is taken to be zero.

If is a principal ideal, then

- .
^{[3]}

The norm is completely multiplicative: if and are ideals of , then

- .
^{[3]}

Thus the absolute norm extends uniquely to a group homomorphism

defined for all nonzero fractional ideals of .

The norm of an ideal can be used to give an upper bound on the field norm of the smallest nonzero element it contains:

there always exists a nonzero for which

where

- is the discriminant of and
- is the number of pairs of (non-real) complex embeddings of
*L*into (the number of complex places of*L*).^{[4]}