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.)
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
is the prime ideal of A lying below
one can equivalently define
to be the fractional ideal of A generated by the set
of field norms of elements of B.
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
, an abuse of notation that is compatible with also writing
for the field norm, as noted above.
, it is reasonable to use positive rational numbers as the range for
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
coincides with the absolute norm defined below.
be a number field with ring of integers
a nonzero (integral) ideal of
The absolute norm of
is By convention, the norm of the zero ideal is taken to be zero.
is a principal ideal, then The norm is completely multiplicative: if
, 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