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.
For , one has , where .
The ideal norm of a principal ideal is thus compatible with the field norm of an element:
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
The norm is completely multiplicative: if and are ideals of , then
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
- 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).