Absolute norm of an ideal

For a nonzero integral ideal $\mathfrak a\subseteq\mathcal O_K$ in the ring of integers of a number field, its absolute norm is

N\mathfrak a=|\mathcal O_K/\mathfrak a|=[\mathcal O_K:\mathfrak a].

The quotient is finite; this index is taken for the additive groups. The unit ideal has norm one.

Examples and distinction

For $K=\mathbb Q$ , $N(n\mathbb Z)=|n|$ when $n\ne0$ . In a degree- $r$ number field, $N(n\mathcal O_K)=|n|^r$ .

The ideal norm takes an ideal as input. The field norm takes an element; they are related by $N(\alpha\mathcal O_K)=|N_{K/\mathbb Q}(\alpha)|$ for nonzero integral $\alpha$ .

References

J. S. Milne, Algebraic Number Theory, v3.08. Author’s text, §4, ideal norm and lattice index.