Let $(L,w)/(K,v)$ be a finite extension of valuation fields, and let $L_w$, $K_v$ be the respective completion of $(L,w)$, $(K.v)$. Is the field extension $L_w/K_v$ finite?

For nonarchemidean valuations on number fields it is trivial, as everything is explicit in terms of prime ideals. For archemidean valuations this is also trivial if we know what they are, but I'm thinking about turning this around and deriving the archemidean valuations on number fields from this (seemingly easy) lemma.

Also, I'm wondering how far can we push it towards nondiscrete, non-Noetherian domains (pun?)

Let $(L,w)/(K,v)$ be a finite extension of valuation fields, and let $L_w$, $K_v$ be the respective completions of $(L,w)$, $(K,v)$. Is the field extension $L_w/K_v$ finite?

For nonarchimedean valuations on number fields it is trivial, as everything is explicit in terms of prime ideals. For archimedean valuations this is also trivial if we know what they are, but I'm thinking about turning this around and deriving the archimedean valuations on number fields from this (seemingly easy) lemma.

Also, I'm wondering how far can we push it towards nondiscrete, non-Noetherian domains (pun?).