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I'm busy working through several books and online notes about Algebraic Number Theory and I came across a question that is very similar to proving something about what Neukirch calls the fundamental identity, that is, as in Neukirch's Algebraic Number Theory Proposition 6.8 Chapter II.

If we assume $L/K$ is a finite extension of degree $n$, with a valuation $v$ on a henselian field $K$ and $w$ is the unique exponential valuation extending $v$ on $L$, we denote the ramification index $$e=e(w/v) = (w(L^{\times}) : v(K^{\times}))$$ and the residue class degree (inertia degree) by $$f=f(w/v)=[l:k]$$ where $k,l$ denote the residue class fields of $K,L$ respectively, then we have that when $v$ is discrete and $L/K$ is separable then we have the fundamental identity $$[L:K] = ef.$$

Firstly, is the assumption that $K$ is henselian necessary? Is the assumption that $K$ being complete with $v$ not enough?

Then, my question is:

Is it true that $e,f$ are finite if and only if $[L:K]$ is finite when the extension is algebraic and not just finite, assuming $K$ is complete with $v$, and $v$ discrete? If so, how do you prove this?

I've found resources stating that this is the case when the extension is finite and separable, but not algebraic.]

Similar: Fundamental Identity - Extension of Valuations

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    $\begingroup$ You ask if being complete is "enough" rather than being Henselian, but completeness is not a consequence of being Henselian: the $p$-adic algebraic numbers (the algebraic numbers inside $\mathbf Q_p$) are a field just like the real algebraic numbers, and they satisfy Hensel's lemma, but the $p$-adic algebraic numbers are not a complete field. $\endgroup$
    – KCd
    Mar 16, 2023 at 17:09

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If you remove the separability condition on $L/K$ and keep the other conditions then always $ef \leq [L:K]$, but equality is no longer necessarily going to occur. When $L/K$ has characteristic $p$, there is an additional positive integer factor called the defect, and denoted $d$, that makes $[L:K] = def$. The defect is a power of $p$ and it is $1$ when $L/K$ is separable, and may or may not be $1$ when $L/K$ is inseparable. When $d = 1$, we say $L/K$ is defectless. A finite extension of $K$ is defectless if it is separable, but finite inseparable extensions of $K$ may or may not be defectless.

The defect factor was discovered by Ostrowski. See Section 5.2 of Roquette's paper on the history of valuation theory here.

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