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 I A and I B 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
N B / A : I B → I A is the unique group homomorphism that satisfies
N B / A ( q ) = p [ B / q : A / p ] for all nonzero prime ideals q of B, where p = q ∩ A is the prime ideal of A lying below q .
Alternatively, for any b ∈ I B one can equivalently define N B / A ( b ) to be the fractional ideal of A generated by the set { N L / K ( x ) | x ∈ b } of field norms of elements of B.
For a ∈ I A , one has N B / A ( a B ) = a n , where n = [ L : K ] . The ideal norm of a principal ideal is thus compatible with the field norm of an element: N B / A ( x B ) = N L / K ( x ) A .
Let L / K be a Galois extension of number fields with rings of integers O K ⊂ O L . Then the preceding applies with A = O K , B = O L , and for any b ∈ I O L we have
N O L / O K ( b ) = O K ∩ ∏ σ ∈ Gal ( L / K ) σ ( b ) , which is an element of I O K . The notation N O L / O K 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 K = Q , it is reasonable to use positive rational numbers as the range for N O L / Z since Z has trivial ideal class group and unit group { ± 1 } , thus each nonzero fractional ideal of Z is generated by a uniquely determined positive rational number. Under this convention the relative norm from L down to K = Q coincides with the absolute norm defined below.
Let L be a number field with ring of integers O L , and a a nonzero (integral) ideal of O L . The absolute norm of a is
N ( a ) := [ O L : a ] = | O L / a | . By convention, the norm of the zero ideal is taken to be zero.
If a = ( a ) is a principal ideal, then N ( a ) = | N L / Q ( a ) | .
The norm is completely multiplicative: if a and b are ideals of O L , then N ( a ⋅ b ) = N ( a ) N ( b ) . Thus the absolute norm extends uniquely to a group homomorphism
N : I O L → Q > 0 × , defined for all nonzero fractional ideals of O L .
The norm of an ideal 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 a ∈ a for which
| N L / Q ( a ) | ≤ ( 2 π ) s | Δ L | N ( a ) , where Δ L is the discriminant of L and s is the number of pairs of (non-real) complex embeddings of L into C (the number of complex places of L).
