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-theory) of a complicated number ring in terms of an ideal-theory) 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. (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 \mathfrak q of B, where is the prime ideal of A lying below \mathfrak q. 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 n = [L : K]. The ideal norm of a principal ideal is thus compatible with the field norm of an element: Let L/K 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 N_{L/K}, an abuse of notation that is compatible with also writing N_{L/K} for the field norm, as noted above. In the case, it is reasonable to use positive rational numbers as the range for since \mathbb{Z} has trivial ideal class group and unit group {\pm 1}, thus each nonzero fractional ideal of \mathbb{Z} is generated by a uniquely determined positive rational number. Under this convention the relative norm from L down to coincides with the absolute norm defined below.

Absolute norm

Let L be a number field with ring of integers, and \mathfrak a a nonzero (integral) ideal of. The absolute norm of \mathfrak a 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 \mathfrak a and \mathfrak b 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 \mathfrak a 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 L into \mathbb{C} (the number of complex places of L ).