Ideal norm


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.

Relative norm

Let A be a Dedekind domain with field of fractions K and integral closure of B in a finite separable extension L of K. Let and be the ideal groups of A and B, respectively 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.

Absolute norm

Let be a number field with ring of integers, and a nonzero 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
where is the discriminant of and is the number of pairs of complex embeddings of into .