A paper I'm reading implicitly assumes the statement: Let $K\_0$ be the completion of $\mathbb {Q}\_ p^{un}$. Then any finite extension of $K\_0$ is complete with residue field $\bar {\mathbb {F}} \_p$. So here are a few questions:
1. Why is this true? Is it true in general that if you complete, and then take algebraic closure - then that is complete?
2. Is it true that any finite extension of $K \_0$ comes from the completion of a finite extension of $\mathbb {Q} \_p ^ {un}$?







Fiddling with p-adics
A finite extension of a complete field is complete. This is standard and proved in lots of places e.g. one of the first two chapters of Cassels-Froehlich, or Bosch-Guentzer-Remmert, or lots of other places. The residue field will only go up a finite amount too, so it must have stayed the same! 


Your Q1 appears to have two parts: the first is "why is this true", which I think I just answered. The second part is not true: if you complete Q at p you get Q\_p and if you then take alg closure you get Q\_p-bar which isn't complete. What is true is that if you take alg closure and then complete, you're still alg closed. This is proved in BGR too. 


Q2 is also true and this will follow from Krasner's lemma, also proved in BGR (and in lots of other places). A finite extension of K\_0 is K\_0(alpha) with alpha a root of a polynomial in K\_0[x]. By Krasner I can perturb the coefficients of this polynomial without changing the extension. So I perturb them until they're in Q\_p^{un} and there's my finite extension of Q\_p^{un}.


