Ideal norm

Let $K\subset L$ be two number fields with ring of integers $R\subset S$. Suppose that the extension $K/L$ is a Galois extension with $G=\textstyle {Gal}(K/L)$. The norm of an ideal $I$ of $S$ is defined as follows

$N_{K}^{L}(I)=R\cap \prod _{\sigma \in G}^{}\sigma (I)$

which is an ideal of $R$. The norm of a principle 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 S to the set of ideals of R. It's 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.

See Also

Dedekind zeta function