September 11th
Today I learned some facts about the different ideal $\diff I,$ where $I$ is a fractional ideal of $\mathcal O_K.$ I think the most interesting is that\[\diff I=I\diff\mathcal O_K,\]Namely, this means we really only care about the global information $\diff\mathcal O_K,$ even though $I^*$ does not look super well-behaved. Without rigor, this is not hard to convince oneself of, for it is equivalent to\[I^*=I^{-1}\mathcal O_K^*\]after taking inverses of everything. (One has to show that $I^*$ and $\diff I$ are fractional ideals first, which is not easy.) Taking a moment for philosophy, we see that the reason we can talk intelligently about $\diff I$ in terms of the entire number ring is that "difficulty'' in the $I^*$ can be translated seamlessly upwards to $\mathcal O_K^*.$ Anyways, the above is equivalent to\[II^*=\mathcal O_K^*.\]Then this we can prove in steps, showing $II^*\subseteq\mathcal O_K^*$ and $I^{-1}\mathcal O_K^*\subseteq I^*,$ which completes the proof. Neither of these lemmas are particularly enlightening. As an aside, taking $I\mapsto I^*$ in the above tells us that $II^*=\mathcal O_K^*=I^*(I^*)^*,$ from which $I=(I^*)^*$ follows.