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

$N_{B/A}(\mathfrak{𝔮})={\mathfrak{𝔭}}^{[B/\mathfrak{𝔮}:A/\mathfrak{𝔭}]}$

for all nonzero prime ideals $\mathfrak{𝔮}$ of B, where $\mathfrak{𝔭}=\mathfrak{𝔮}\cap A$ is the prime ideal of A lying below $\mathfrak{𝔮}$.

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

For $\mathfrak{𝔞}\in {{ℐ}}_A$, one has $N_{B/A}(\mathfrak{𝔞}B)={\mathfrak{𝔞}}^n$, 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 $\mathfrak{𝔟}\in {{ℐ}}_{{{𝒪}}_L}$ we have

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

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 $K=\mathbb{\mathbb{Q}}$, it is reasonable to use positive rational numbers as the range for $N_{{{𝒪}}_L/\mathbb{\mathbb{Z}}}$ since $\mathbb{\mathbb{Z}}$ has trivial ideal class group and unit group $\{\pm 1\}$, thus each nonzero fractional ideal of $\mathbb{\mathbb{Z}}$ is generated by a uniquely determined positive rational number. Under this convention the relative norm from $L$ down to $K=\mathbb{\mathbb{Q}}$ coincides with the absolute norm defined below.

Absolute norm

Let $L$ be a number field with ring of integers ${{𝒪}}_L$, and $\mathfrak{𝔞}$ a nonzero (integral) ideal of ${{𝒪}}_L$.

The absolute norm of $\mathfrak{𝔞}$ is

$N(\mathfrak{𝔞}):=[{{𝒪}}_L:\mathfrak{𝔞}]=|{{𝒪}}_L/\mathfrak{𝔞}|.$

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

If $\mathfrak{𝔞}=(a)$ is a principal ideal, then

$N(\mathfrak{𝔞})=|N_{L/\mathbb{\mathbb{Q}}}(a)|$.

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

$N(\mathfrak{𝔞}\cdot \mathfrak{𝔟})=N(\mathfrak{𝔞})N(\mathfrak{𝔟})$.

Thus the absolute norm extends uniquely to a group homomorphism

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

defined for all nonzero fractional ideals of ${{𝒪}}_L$.

The norm of an ideal $\mathfrak{𝔞}$ can be used to give an upper bound on the field norm of the smallest nonzero element it contains:

there always exists a nonzero $a\in \mathfrak{𝔞}$ for which

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

where

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

See also

• Field norm
• Dedekind zeta function