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.

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 ${ℐ}_A$ 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

$N_{B/A}:{ℐ}_B\to {ℐ}_A$

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 $𝔭=𝔮\cap A$ is the prime ideal of A lying below $𝔮$.

Alternatively, for any $𝔟\in {ℐ}_B$ one can equivalently define $N_{B/A}(𝔟)$ to be the fractional ideal of A generated by the set $\{N_{L/K}(x)|x\in 𝔟\}$ of field norms of elements of B.

For $𝔞\in {ℐ}_A$, one has $N_{B/A}(𝔞B)={𝔞}^n$, where $n=[L:K]$.

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

$N_{B/A}(xB)=N_{L/K}(x)A.$

Let $L/K$ be a Galois extension of number fields with rings of integers ${𝒪}_K\subset {𝒪}_L$.

Then the preceding applies with $A={𝒪}_K,B={𝒪}_L$, and for any $𝔟\in {ℐ}_{{𝒪}_L}$ we have

$N_{{𝒪}_L/{𝒪}_K}(𝔟)=K\cap \prod _{\sigma \in \mathrm{Gal}(L/K)}\sigma (𝔟),$

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 $K=\mathbb{Q}$, it is reasonable to use positive rational numbers as the range for $N_{{𝒪}_L/\mathbb{Z}}$ 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 $K=\mathbb{Q}$ coincides with the absolute norm defined below.

Absolute norm

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

$N(𝔞)=|N_{L/\mathbb{Q}}(a)|$.

The norm is completely multiplicative: if $𝔞$ and $𝔟$ are ideals of ${𝒪}_L$, then

$N(𝔞\cdot 𝔟)=N(𝔞)N(𝔟)$.

Thus the absolute norm extends uniquely to a group homomorphism

$N:{ℐ}_{{𝒪}_L}\to {\mathbb{Q}}_{>0}^{\times },$

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 $a\in 𝔞$ for which

$|N_{L/\mathbb{Q}}(a)|\le {\left(\frac{2}{\pi }\right)}^s\sqrt{\left|{\mathrm{\Delta }}_L\right|}N(𝔞),$

where

• ${\mathrm{\Delta }}_L$ is the discriminant of $L$ and
• $s$ is the number of pairs of (non-real) complex embeddings of Template:Math into $ℂ$ (the number of complex places of Template:Math).

See also

• Field norm
• Dedekind zeta function