Ideal norm explained

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.

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

l{I}_A

and

l{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\colonl{I}_B\tol{I}_A

is the unique group homomorphism that satisfies

N_B/A(akq)=ak{p}^[B/akq

for all nonzero prime ideals

akq

of B, where

akp=akq\capA

is the prime ideal of A lying below

akq

Alternatively, for any

akb\inl{I}_B

one can equivalently define

N_B/A(ak{b})

to be the fractional ideal of A generated by the set

\{N_L/K(x)|x\inak{b}\}

of field norms of elements of B.

For

aka\inl{I}_A

, one has

N_B/A(akaB)=akaⁿ

, where

n=[L:K]

The ideal norm of a principal ideal is thus compatible with the field norm of an element:

N_B/A(xB)=N_L/K(x)A.

Let

L/K

be a Galois extension of number fields with rings of integers

l{O}_K\subsetl{O}_L

Then the preceding applies with

A=l{O}_K,B=l{O}_L

, and for any

akb\inl{I}_l{O_L}

we have

N_l{O_L/l{O}_K}(akb)=K\cap\prod_\sigma(L/K)}\sigma(akb),

which is an element of

l{I}_l{O_K}

The notation

N_l{O_L/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_l{O_L/Z

}\, since

has trivial ideal class group and unit group

\{\pm1\}

, thus each nonzero fractional ideal of

is generated by a uniquely determined positive rational number.Under this convention the relative norm from

down to

K=Q

coincides with the absolute norm defined below.

Absolute norm

Let

be a number field with ring of integers

l{O}_L

, and

aka

a nonzero (integral) ideal of

l{O}_L

The absolute norm of

aka

N(aka):=\left[l{O}_L:aka\right]=\left|l{O}_{L/aka\right|.}

By convention, the norm of the zero ideal is taken to be zero.

aka=(a)

is a principal ideal, then

N(aka)=\left|N_L/Q(a)\right|

The norm is completely multiplicative: if

aka

and

akb

are ideals of

l{O}_L

, then

N(aka ⋅ akb)=N(aka)N(akb)

Thus the absolute norm extends uniquely to a group homomorphism

N\colonl{I}_l{O_L}\toQ

	x ,

	>0

defined for all nonzero fractional ideals of

l{O}_L

aka

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\inaka

for which

\left|N_L/Q(a)\right|\leq\left(

	2
	\pi

\right)^s\sqrt{\left|\Delta_{L\right|}N(aka),}

where

\Delta_L

is the discriminant of

and

is the number of pairs of (non-real) complex embeddings of into

(the number of complex places of).

Ideal norm explained

Relative norm

Absolute norm

See also