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 $\mathcal_A$ and $\mathcal_B$ 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
:$N_\colon \mathcal_B \to \mathcal_A$
is the unique group homomorphism that satisfies
:$N_(\mathfrak q) = \mathfrak^$
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 $\mathfrak b\in\mathcal_B$ one can equivalently define $N_(\mathfrak)$ to be the fractional ideal of ''A'' generated by the set $\$ of field norms of elements of ''B''.

For $\mathfrak a \in \mathcal_A$, one has $N_(\mathfrak a B) = \mathfrak a^n$, where n = : K/math>.

The ideal norm of a principal ideal is thus compatible with the field norm of an element:
:$N_(xB) = N_(x)A.$

Let $L/K$ be a Galois extension of number fields with rings of integers $\mathcal_K\subset \mathcal_L$.

Then the preceding applies with $A = \mathcal_K, B = \mathcal_L$, and for any $\mathfrak b\in\mathcal_$ we have
:$N_(\mathfrak b)= K \cap\prod_ \sigma (\mathfrak b),$
which is an element of $\mathcal_$.

The notation $N_$ is sometimes shortened to $N_$, an abuse of notation that is compatible with also writing $N_$ for the field norm, as noted above.

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

Absolute norm

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

The absolute norm of $\mathfrak a$ is
:N(\mathfrak a) :=\left \mathcal_L: \mathfrak a\right \left, \mathcal_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)=\left, N_(a)\$.

The norm is completely multiplicative: if $\mathfrak a$ and $\mathfrak b$ are ideals of $\mathcal_L$, then

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

Thus the absolute norm extends uniquely to a group homomorphism
:$N\colon\mathcal_\to\mathbb_^\times,$
defined for all nonzero fractional ideals of $\mathcal_L$.

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
:$\left, N_(a)\\leq \left ( \frac\right )^s \sqrtN(\mathfrak a),$
where

:* $\Delta_L$ is the discriminant of $L$ and
:* $s$ is the number of pairs of (non-real) complex embeddings of into $\mathbb$ (the number of complex places of ).

See also

* Field norm
* Dedekind zeta function

References

{{reflist

Algebraic number theory
Commutative algebra
Ideals (ring theory)