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 .
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 .
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
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