Norm of an ideal

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The norm of an ideal is defined in algebraic number theory. Let $K\subset L$ be two number fields with rings of integers $O_K\subset O_L$ . Suppose that the extension $L / K$ is a Galois extension with

$G=\textstyle{Gal}(L/K).$

The norm of an ideal $I$ of $O L$ is defined as follows

$N_K^L(I)=O_K \cap\prod_{\sigma \in G}^{} \sigma (I)$

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 the norm map

$N_\mathbb{Q}^L(I)$

since Z is a principal ideal domain. This idea doesn't work in general since class group is usually non-trivial.

[edit] Alternate Formulation

Let $L$ be a number field with ring of integers $O L$ , and $α$ a nonzero ideal of $O L$ . Then the norm of $α$ 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 $α$ is a principal ideal with $α = (a)$ , then $N (α) = | N (a) |$ .

The norm is also completely multiplicative in that if $α$ and $β$ are ideals of $O L$ , then $N (β) = N (α) N (β)$ .

The norm of an ideal $α$ 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 $Δ L$ is the discriminant of $L$ and $r 2$ is the number of pairs of complex embeddings of $L$ into $\mathbb{C}$ .

[edit] See also

Dedekind zeta function

[edit] References

Daniel A. Marcus, Number Fields, third edition, Springer-Verlag, 1977

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