Norm of a fractional ideal ($N_{L/K}(I)$)

For a fractional ideal $I$ of ${\mathcal{O}}_L$, the relative ideal norm is the fractional ideal of ${\mathcal{O}}_K$ obtained multiplicatively from prime-ideal norms. Equivalently, its local valuation is the determinant valuation of $I$ as an ${\mathcal{O}}_K$-lattice relative to ${\mathcal{O}}_L$.

For a nonzero fractional ideal of a number field, the displayed generated ideal equals the relative norm of a fractional ideal. Locally, multiplication by $x$ on the integral lattice has determinant $N_{L/K}(x)$. A Smith normal form calculation and the residue filtrations give $v_{\mathfrak{p}}(N(x))={\sum }_{\mathfrak{P}\mid \mathfrak{p}}{f}_{\mathfrak{P}/\mathfrak{p}}{v}_{\mathfrak{P}}(x)$. The Chinese remainder theorem for ideals chooses $x\in I$ simultaneously attaining the minimum valuation of $I$ at each prime above a fixed $\mathfrak{p}$. Thus the generated ideal has valuation $\sum f_{\mathfrak{P}/\mathfrak{p}}{v}_{\mathfrak{P}}(I)$. Adding these valuations proves multiplicativity, including fractional ideals after clearing denominators.