= Relative ideal norm from norms of elements
{title2=$N_{L/K}(I)=\langle N_{L/K}(x):x\in I\rangle$}
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.
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