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.^[1]

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: $N_{B/A}(xB) = N_{L/K}(x)A.$^[2]

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

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

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 $N(\mathfrak a)=\left|N_{L/\mathbb{Q}}(a)\right|$.^[3]

The norm is completely multiplicative: if $\mathfrak a$ and $\mathfrak b$ are ideals of $\mathcal{O}_L$, then $N(\mathfrak a\cdot\mathfrak b)=N(\mathfrak a)N(\mathfrak b)$.^[3] 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).^[4]

See also

• Field norm
• Dedekind zeta function

References

1. Janusz, Gerald J. (1996), Algebraic number fields, Graduate Studies in Mathematics 7 (second ed.), Providence, Rhode Island: American Mathematical Society, Proposition I.8.2, ISBN 0-8218-0429-4, MR 1362545 (96j:11137)
2. Serre, Jean-Pierre (1979), Local fields, Graduate Texts in Mathematics 67, Translated from the French by Marvin Jay Greenberg, New York: Springer-Verlag, 1.5, Proposition 14, ISBN 0-387-90424-7, MR 554237 (82e:12016)
3. Marcus, Daniel A. (1977), Number fields, Universitext, New York: Springer-Verlag, Theorem 22c, ISBN 0-387-90279-1, MR 0457396 (56 #15601)
4. Neukirch, Jürgen (1999), Algebraic number theory, Berlin: Springer-Verlag, Lemma 6.2, ISBN 3-540-65399-6, MR 1697859 (2000m:11104)