Indices of inseparability (reviewed)

Let $L/K$ be a finite extension of $p$-adic fields and let $K'/K$ be the maximum unramified subextension of $L/K$. Set $n=[L:K']$ and write $n=up^{\nu}$ with $p\nmid u$. Let $\pi_L$ be a uniformizer for $L$ and let $f(X)=X^n+c_{n-1}X^{n-1}\cdots+c_1X+c_0$ be the minimal polynomial of $\pi_L$ over $K'$. For $k\in\Z$ define $\overline{v}_p(k)=\min\{v_p(k),\nu\}$. For $0\le j\le\nu$ set $i_j^{\pi_L}=\min\{nv_{K'}(c_h)+h-n: 0\le h\le n-1,\;\overline{v}_p(h)\le j\},$ or let $i_j^{\pi_L}=\infty$ if the set above is empty. Heiermann [10.1006/JNTH.1996.0092] defines the $j$th index of inseparability of $L/K$ to be $i_j=\min\{i_{j'}^{\pi_L}+(j'-j)e_L:j\le j'\le\nu\},$ where $e_L=v_L(p)$ is the absolute ramification index of $L$.

It follows that $0=i_{\nu}<i_{\nu-1}\le\dots\le i_1\le i_0$. The value of $i_j^{\pi_L}$ can depend on the choice of $\pi_L$, but $i_j$ depends only on $L/K$. The fact that $i_j$ is well-defined puts constraints on the possibilities for the valuations of the coefficients in the minimal polynomial over $K'$ of any uniformizer for $L$. The indices of inseparability determine the usual ramification data of $L/K$ (e.g., the slopes), and in some cases give new information.