Let $K$ be a number field, let $\mathcal{O}_K$ be its ring of integers, and let $B = \{b_1,\ldots,b_d\}$ be a subset of $K$ of cardinality $d$ such that $\mathcal{O}_K = \mathbb{Z}b_1+\cdots+\mathbb{Z}b_d$. Define the trace-dual basis $B^+$ by the property $\mathrm{Tr}(b_ib'_j)=\delta_{i,j}$ (Kronecker delta).
i) Show that the $\mathbb{Z}$-span $\mathbb{Z}b'_1+\cdots+ \mathbb{Z}b'_d$ of $B^+$ does not depend on the choice of $B$. This abelian group is called the trace dual of $O_K$, and we denote it by $O^+_K$.
ii) Show that $\mathcal{O}^+_K$ contains $\mathcal{O}_K$, and that the discriminant $\Delta(K)$ is the index $[\mathcal{O}^+_K : O_K]$.