Recall the notion of a different of a number field. For a lattice $L$ in a number field $K$, we may define its dual using the trace $\operatorname{Tr}_{K/\mathbf{Q}}$: 

\[L^\vee = \{\alpha \in K: \operatorname{Tr}_{K/\mathbf{Q}}(\alpha L) \in \mathbf{Z}\}.\]

It is easy to verify that a fractional ideal $\mathfrak{a}$ gives a lattice in $K$; it turns out that $\mathfrak{a}^\vee$ is also a fractional ideal and satisfies:

\[\mathfrak{a}^\vee = \mathfrak{a}^{-1}\mathcal{O}_K^\vee.\]

The ideal $\mathcal{O}_K$ is a full sublattice of $K$ and $\mathcal{O}_K^\vee \subseteq \mathcal{O}_K$. Thus, the dual lattice gives us an interesting fractional ideal: the dual $\mathcal{O}_K^\vee$ is the largest fractional ideal whose all elements have integer trace. The inverse $(\mathcal{O}_K^\vee)^{-1}$, therefore an integral ideal, is called the different of the number field $K$.