Ideal norm

Remarks

Example: in $\Z[i]$, the ideal $(1 + i)$ has the classes of $0$ and $1$, so $N((1 + i)) = 2$, which equals the norm of the Gaussian integer $1 + i$, namely $1^2 + 1^2$. Non-example: the zero ideal is excluded, since $\mathcal{O}_K / (0) \cong \mathcal{O}_K$ is infinite. The norm is multiplicative: $N(\mathfrak{a}\mathfrak{b}) = N(\mathfrak{a}) N(\mathfrak{b})$. For a nonzero $\alpha \in \mathcal{O}_K$, $N(\alpha \mathcal{O}_K) = \abs{N_{K/\Q}(\alpha)}$, the absolute value of the field norm. By multiplicativity the norm extends to a group homomorphism from the nonzero fractional ideals of $\mathcal{O}_K$ to $\Q_{>0}^{\times}$. For a nonzero prime ideal $\mathfrak{p}$ lying over the prime $p$, $N(\mathfrak{p}) = p^f$, where $f$ is the residue degree.