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 $$\mathcal{I}_A$$ 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 $L$).