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. 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.