= Trace dual of a fractional ideal
{title2=$\mathfrak b^\vee=\mathfrak D_{K/\mathbb Q}^{-1}\mathfrak b^{-1}$}
For a nonzero <fractional ideal> $\mathfrak b$ of a <number field>, its <trace-dual lattice> is the <fractional ideal>
$$
\mathfrak b^\vee=\{z\in K:\operatorname{Tr}_{K/\mathbb Q}(z\mathfrak b)\subseteq\mathbb Z\}=\mathfrak D_{K/\mathbb Q}^{-1}\mathfrak b^{-1}.
$$
The equality follows from the definition of the <inverse different> and invertibility of <fractional ideals>. Under the positive metric on the <Minkowski embedding of a number field>, the Euclidean <dual lattice> of $j(\mathfrak b)$ is $\overline{j(\mathfrak b^\vee)}$: coordinatewise <complex conjugation> converts the positive <inner product> into the <trace pairing>. This assertion does not require that conjugation preserve the number field itself.
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