September 14th
Today I learned a more geometric interpretation of the $A^*$ in the definition of the different ideal. As usual, fix a finite extension $L/K$ of degree $n,$ and let $A$ be generated by $\{\alpha_1,\ldots,\alpha_n\}.$ Mathematics says that the natural thing is not to focus on the particular elements in $L/K$ but also put all the embeddings on the same fitting. So it is natural to organize things into\[{\bf A}=\begin{bmatrix} \sigma_1(\alpha_1) & \sigma_2(\alpha_1) & \cdots & \sigma_n(\alpha_1) \\ \sigma_1(\alpha_2) & \sigma_2(\alpha_2) & \cdots & \sigma_n(\alpha_2) \\ \vdots & \vdots & \ddots & \vdots \\ \sigma_1(\alpha_n) & \sigma_2(\alpha_n) & \cdots & \sigma_n(\alpha_n)\end{bmatrix}.\]This matrix, naturally, defines a lattice. Then $A^*$ is associated with the $\textit{dual lattice}$ of $A,$ generated by ${\bf A}^{-\intercal}.$ One can show that the this matrix takes form\[{\bf A}^{-\intercal}=\begin{bmatrix} \sigma_1(\beta_1) & \sigma_2(\beta_1) & \cdots & \sigma_n(\beta_1) \\ \sigma_1(\beta_2) & \sigma_2(\beta_2) & \cdots & \sigma_n(\beta_2) \\ \vdots & \vdots & \ddots & \vdots \\ \sigma_1(\beta_n) & \sigma_2(\beta_n) & \cdots & \sigma_n(\beta_n)\end{bmatrix}.\]The inverse, of course, exists. There is some technicality showing that the rows are generated by single $\beta_k,$ but we can show this by considering what happens when applying some $\sigma_\ell\sigma_k^{-1}.$
It happens that these $\{\beta_1,\ldots,\beta_n\}$ generate $A^*.$ To provide some reasoning why, the reason the trace appears at all is that\[\begin{bmatrix} \sigma_1(\alpha_1) & \cdots & \sigma_n(\alpha_1) \\ \vdots & \ddots & \vdots \\ \sigma_1(\alpha_n) & \cdots & \sigma_n(\alpha_n)\end{bmatrix}\begin{bmatrix} \sigma_1(\beta_1) & \cdots & \sigma_1(\beta_n) \\ \vdots & \ddots & \vdots \\ \sigma_n(\beta_1) & \cdots & \sigma_n(\beta_n)\end{bmatrix}=\begin{bmatrix} \op T_K^L(\alpha_1\beta_1) & \cdots & \op T_K^L(\alpha_1\beta_n) \\ \vdots & \ddots & \vdots \\ \op T_K^L(\alpha_n\beta_1) & \cdots & \op T_K^L(\alpha_n\beta_n)\end{bmatrix},\]so we get some trace information encoded by our definition of the lattice dual. There's a nice picture of the dual lattice that I don't really fully understand, but it's comforting to know that $A^*$ isn't coming out purely nowhere. For example, the determinant of the lattice is inverted, so there is some notion that $\diff I=(I^*)^{-1}$ is taking "incomptabile'' inverses to create an object that should still encode some information about $I.$