# Finiteness of degree of algebraic extension if and only if ramification and residue class degrees are finite

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.]

• 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.
– KCd
Mar 16, 2023 at 17:09

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.