Krasner's lemma states that if 
a complete field with valuation 
, 
is a fixed algebraic closure of 
together with the canonical extension of 
, and 
is its completion with respect to 
, then 
remains algebraically closed.
Krasner's Lemma
See also
Non-Archimedean ValuationThis entry contributed by José Gallardo Alberni
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Alberni, José Gallardo. "Krasner's Lemma." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/KrasnersLemma.html