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 [math]\displaystyle{ \mathcal{I}_A }[/math] and [math]\displaystyle{ \mathcal{I}_B }[/math] 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

[math]\displaystyle{ N_{B/A}\colon \mathcal{I}_B \to \mathcal{I}_A }[/math]

is the unique group homomorphism that satisfies

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

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

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

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

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

[math]\displaystyle{ N_{B/A}(xB) = N_{L/K}(x)A. }[/math]

Let [math]\displaystyle{ L/K }[/math] be a Galois extension of number fields with rings of integers [math]\displaystyle{ \mathcal{O}_K\subset \mathcal{O}_L }[/math].

Then the preceding applies with [math]\displaystyle{ A = \mathcal{O}_K, B = \mathcal{O}_L }[/math], and for any [math]\displaystyle{ \mathfrak b\in\mathcal{I}_{\mathcal{O}_L} }[/math] we have

[math]\displaystyle{ N_{\mathcal{O}_L/\mathcal{O}_K}(\mathfrak b)= K \cap\prod_{\sigma \in \operatorname{Gal}(L/K)} \sigma (\mathfrak b), }[/math]

which is an element of [math]\displaystyle{ \mathcal{I}_{\mathcal{O}_K} }[/math].

The notation [math]\displaystyle{ N_{\mathcal{O}_L/\mathcal{O}_K} }[/math] is sometimes shortened to [math]\displaystyle{ N_{L/K} }[/math], an abuse of notation that is compatible with also writing [math]\displaystyle{ N_{L/K} }[/math] for the field norm, as noted above.

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

Absolute norm

Let [math]\displaystyle{ L }[/math] be a number field with ring of integers [math]\displaystyle{ \mathcal{O}_L }[/math], and [math]\displaystyle{ \mathfrak a }[/math] a nonzero (integral) ideal of [math]\displaystyle{ \mathcal{O}_L }[/math].

The absolute norm of [math]\displaystyle{ \mathfrak a }[/math] is

[math]\displaystyle{ N(\mathfrak a) :=\left [ \mathcal{O}_L: \mathfrak a\right ]=\left|\mathcal{O}_L/\mathfrak a\right|.\, }[/math]

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

If [math]\displaystyle{ \mathfrak a=(a) }[/math] is a principal ideal, then

[math]\displaystyle{ N(\mathfrak a)=\left|N_{L/\mathbb{Q}}(a)\right| }[/math].

The norm is completely multiplicative: if [math]\displaystyle{ \mathfrak a }[/math] and [math]\displaystyle{ \mathfrak b }[/math] are ideals of [math]\displaystyle{ \mathcal{O}_L }[/math], then

[math]\displaystyle{ N(\mathfrak a\cdot\mathfrak b)=N(\mathfrak a)N(\mathfrak b) }[/math].

Thus the absolute norm extends uniquely to a group homomorphism

[math]\displaystyle{ N\colon\mathcal{I}_{\mathcal{O}_L}\to\mathbb{Q}_{\gt 0}^\times, }[/math]

defined for all nonzero fractional ideals of [math]\displaystyle{ \mathcal{O}_L }[/math].

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

there always exists a nonzero [math]\displaystyle{ a\in\mathfrak a }[/math] for which

[math]\displaystyle{ \left|N_{L/\mathbb{Q}}(a)\right|\leq \left ( \frac{2}{\pi}\right )^s \sqrt{\left|\Delta_L\right|}N(\mathfrak a), }[/math]

where

• [math]\displaystyle{ \Delta_L }[/math] is the discriminant of [math]\displaystyle{ L }[/math] and
• [math]\displaystyle{ s }[/math] is the number of pairs of (non-real) complex embeddings of L into [math]\displaystyle{ \mathbb{C} }[/math] (the number of complex places of L).

See also

• Field norm
• Dedekind zeta function