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 ${\mathcal {I))_{A)$ and ${\mathcal {I))_{B)$ 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

$N_{B/A}\colon {\mathcal {I))_{B}\to {\mathcal {I))_{A)$ 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 ${\mathfrak {b))\in {\mathcal {I))_{B)$ 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 ${\mathfrak {a))\in {\mathcal {I))_{A)$ , 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 ${\mathcal {O))_{K}\subset {\mathcal {O))_{L)$ .

Then the preceding applies with $A={\mathcal {O))_{K},B={\mathcal {O))_{L)$ , and for any ${\mathfrak {b))\in {\mathcal {I))_((\mathcal {O))_{L))$ we have

$N_((\mathcal {O))_{L}/{\mathcal {O))_{K))({\mathfrak {b)))=K\cap \prod _{\sigma \in \operatorname {Gal} (L/K)}\sigma ({\mathfrak {b))),$ which is an element of ${\mathcal {I))_((\mathcal {O))_{K))$ .

The notation $N_((\mathcal {O))_{L}/{\mathcal {O))_{K))$ 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 $N_((\mathcal {O))_{L}/\mathbb {Z} }\,$ 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 ${\mathcal {O))_{L)$ , and ${\mathfrak {a))$ a nonzero (integral) ideal of ${\mathcal {O))_{L)$ .

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

$N({\mathfrak {a))):=\left[{\mathcal {O))_{L}:{\mathfrak {a))\right]=\left|{\mathcal {O))_{L}/{\mathfrak {a))\right|.\,$ By convention, the norm of the zero ideal is taken to be zero.

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

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

The norm is completely multiplicative: if ${\mathfrak {a))$ and ${\mathfrak {b))$ are ideals of ${\mathcal {O))_{L)$ , then

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

Thus the absolute norm extends uniquely to a group homomorphism

$N\colon {\mathcal {I))_((\mathcal {O))_{L))\to \mathbb {Q} _{>0}^{\times },$ defined for all nonzero fractional ideals of ${\mathcal {O))_{L)$ .

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

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

• $\Delta _{L)$ 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).