Assume I am given a finite primitive extension of complete discretely valued fields $L=K(\alpha)/K$, say with monic integral minimal polynomial $f$ for $\alpha$.
How does one systematically compute the ramification and inertia indices $e$ and $f$?
Do I know that the ring of integers $A_L$ is automatically generated over $A_K$ by $\alpha$? If so, then I can reduce $f$ to obtain the extension of residue fields, and compute the inertia that way. I know that such a generator exists, but it isn’t clear the $\alpha$ I started with works.
I am mostly concerned with finite extensions of the $p$-adics. It seems like this local theory should be computationally straightforward, but a systematic procedure for computing the ramification eludes me.
Ramification is known if you know a uniformizer, or know the valuation on $L$ that extends the valuation on $K$, but both of these require the ramification index to compute.
In the Galois case, the filtration by higher ramification groups gives some restriction on what subgroups of $G$ can be the inertia group, but this data is insufficient in general.
I know that I can express $L$ as adjoining a root of an Eisenstein polynomial if and only if the extension is totally ramified. But this feels like a limited trick, as the substitutions required to get an Eisenstein polynomial (and thus a uniformizer upstairs) are not usually obvious. More generally, the Newton polygon can give you some bounds on ramification, but I don’t see how it can tell you the exact value of ramification in general.