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 the field of fractions K and B be the integral closure of A in a finite separable extension L of K. (In particular, B is Dedekind then.) Let $\operatorname{Id}(A)$ and $\operatorname{Id}(B)$ be the ideal groups of A and B, respectively (i.e., the sets of fractional ideals.) Following (Serre 1979), the norm map

$N_{B/A}: \operatorname{Id}(B) \to \operatorname{Id}(A)$

is a homomorphism given by

$N_{B/A}(\mathfrak q) = \mathfrak{p}^{[B/\mathfrak q : A/\mathfrak p]}, \mathfrak q \in \operatorname{Spec} B, \mathfrak q | \mathfrak p.$

If $L, K$ are local fields, $N_{B/A}(\mathfrak{b})$ is defined to be a fractional ideal generated by the set $\{ N_{L/K}(x) | x \in \mathfrak{b} \}.$ This definition is equivalent to the above and is given in (Iwasawa 1986).

For $\mathfrak a \in \operatorname{Id}(A)$, one has $N_{B/A}(\mathfrak a B) = \mathfrak a^n$ where $n = [L : K]$. The definition is thus also compatible with norm of an element: $N_{B/A}(xB) = N_{L/K}(x)A.$

Let $L/K$ be a finite Galois extension of number fields with rings of integer $\mathcal{O}_K\subset \mathcal{O}_L$. Then the preceding applies with $A = \mathcal{O}_K, B = \mathcal{O}_L$ and one has

$N_{L/K}(I)=\mathcal{O}_K \cap\prod_{\sigma \in G}^{} \sigma (I)\,$

which is an ideal of $\mathcal{O}_K$. The norm of a principal ideal generated by α is the ideal generated by the field norm of α.

The norm map is defined from the set of ideals of $\mathcal{O}_L$ to the set of ideals of $\mathcal{O}_K$. It is reasonable to use integers as the range for $N_{L/\mathbf{Q}}\,$ since Z has trivial ideal class group. This idea does not work in general since the class group may not be trivial.

Absolute norm

Let $L$ be a number field with ring of integers $\mathcal{O}_L$, and $\alpha$ a nonzero ideal of $\mathcal{O}_L$. Then the norm of $\alpha$ is defined to be

$N(\alpha) =\left [ \mathcal{O}_L: \alpha\right ]=|\mathcal{O}_L/\alpha|.\,$

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

If $\alpha$ is a principal ideal with $\alpha=(a)$, then $N(\alpha)=|N(a)|$. For proof, cf. Marcus, theorem 22c, pp65ff.

The norm is also completely multiplicative in that if $\alpha$ and $\beta$ are ideals of $\mathcal{O}_L$, then $N(\alpha\cdot\beta)=N(\alpha)N(\beta)$. For proof, cf. Marcus, theorem 22a, pp65ff.

The norm of an ideal $\alpha$ can be used to bound the norm of some nonzero element $x\in \alpha$ by the inequality

$|N(x)|\leq \left ( \frac{2}{\pi}\right ) ^ {r_2} \sqrt{|\Delta_L|}N(\alpha)$

where $\Delta_L$ is the discriminant of $L$ and $r_2$ is the number of pairs of complex embeddings of $L$ into $\mathbf{C}$.

See also

• Dedekind zeta function

References

• Iwasawa, Kenkichi (1986), Local class field theory, Oxford Science Publications, New York: The Clarendon Press Oxford University Press, pp. viii+155, ISBN 0-19-504030-9, MR 863740 (88b:11080) 
• Marcus, Daniel A. (1977), Number fields, New York: Springer-Verlag, pp. viii+279, ISBN 0-387-90279-1, MR 0457396 (56 #15601) 
• Serre, Jean-Pierre (1979), Local fields, Graduate Texts in Mathematics 67, New York: Springer-Verlag, pp. viii+241, ISBN 0-387-90424-7, MR 554237 (82e:12016)