= Norm of the different ideal
{title2=$\mathfrak d_{E/K}=N_{E/K}(\mathfrak D_{E/K})$}
The <relative discriminant> of a finite extension of <number fields> is the relative <norm of a fractional ideal> applied to its <different ideal>. Consequently, at $\mathfrak p$,
$$
v_{\mathfrak p}(\mathfrak d_{E/K})=\sum_{\mathfrak P\mid\mathfrak p}f_{\mathfrak P/\mathfrak p}d_{\mathfrak P},
$$
where $f_{\mathfrak P/\mathfrak p}$ is the <residue-field degree> and $d_{\mathfrak P}$ is the <different exponent>. Over $\mathbb Q$, $N(\mathfrak D_{E/\mathbb Q})=|D_E|$.
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