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 the field of fractions K and B be the integral closure of A in a finite separable extension L of K. (In particular, B is Dedekind then.) Let and be the ideal groups of A and B, respectively (i.e., the sets of fractional ideals.) Following (Serre 1979), the norm map
is a homomorphism given by
If are local fields, is defined to be a fractional ideal generated by the set This definition is equivalent to the above and is given in (Iwasawa 1986).
For , one has where . The definition is thus also compatible with norm of an element: 
Let be a finite Galois extension of number fields with rings of integer . Then the preceding applies with and one has
The norm map is defined from the set of ideals of to the set of ideals of . It is reasonable to use integers as the range for since Z has trivial ideal class group. This idea does not work in general since the class group may not be trivial.
Let be a number field with ring of integers , and a nonzero ideal of . Then the norm of is defined to be
By convention, the norm of the zero ideal is taken to be zero.
If is a principal ideal with , then . For proof, cf. Marcus, theorem 22c, pp65ff.
The norm is also completely multiplicative in that if and are ideals of , then . For proof, cf. Marcus, theorem 22a, pp65ff.
The norm of an ideal can be used to bound the norm of some nonzero element by the inequality
where is the discriminant of and is the number of pairs of complex embeddings of into .
- Serre, 1. 5, Proposition 14.
- Iwasawa, Kenkichi (1986), Local class field theory, Oxford Science Publications, New York: The Clarendon Press Oxford University Press, pp. viii+155, ISBN 0-19-504030-9, MR 863740 (88b:11080) Unknown parameter
- Marcus, Daniel A. (1977), Number fields, New York: Springer-Verlag, pp. viii+279, ISBN 0-387-90279-1, MR 0457396 (56 #15601) Unknown parameter
- Serre, Jean-Pierre (1979), Local fields, Graduate Texts in Mathematics 67, New York: Springer-Verlag, pp. viii+241, ISBN 0-387-90424-7, MR 554237 (82e:12016) Unknown parameter