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.

1 Relative norm
2 Absolute norm
3 See also
4 References

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 also compatible with norm of an element: $N_{B/A}(xB) = N_{L/K}(x)A.$ ^[1]

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

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

which is an ideal of $O_K$ . 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 $O_L$ to the set of ideals of $O_K$ . It is reasonable to use integers as the range for $N_{L/\mathbb{Q}}\,$ 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 $O_L$ , and $\alpha$ a nonzero ideal of $O_L$ . Then the norm of $\alpha$ is defined to be

$N(\alpha) =\left [ O_L: \alpha\right ]=|O_L/\alpha|.\,$

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

If $\alpha$ is a principal ideal with $\alpha=(a)$ , then $N(\alpha)=|N(a)|$ .

The norm is also completely multiplicative in that if $\alpha$ and $\beta$ are ideals of $O_L$ , then $N(\alpha*\beta)=N(\alpha)N(\beta)$ .

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 $\mathbb{C}$ .

References

^ Serre, 1. 5, Proposition 14.

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 (88b:11080) 863740 (88b:11080)
Marcus, Daniel A. (1977), Number fields, New York: Springer-Verlag, pp. viii+279, ISBN 0-387-90279-1, MR (56 #15601) 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 (82e:12016) 554237 (82e:12016)

Ideal norm

Contents

Relative norm

Absolute norm

See also

References