I have been working with p-adically closed fields and there are two results that are used time and times again in what I am reading, but I cannot find any references where they are proved...


The first is that p-adically closed fields have a finite number of algebraic extensions of a given degree, the other that all algebraic extensions of a p-adically closed field are generated by an element that is algebraic over Q.


As there is (it seems to me) a lot more literature on p-adic fields than on p-adically closed fields, if anyone had an idea where these results might be proved for Qp, I would be very grateful (although if someone had a reference for p-adically closed fields directly I do not mind).


(added on the 17th of July)


I have been wondering lately if there is a way, when one has a finite extension $K$ of $Q\_p$ of finding a generator $a$ of $K$ that is algebraic over $Q$ that is also a generator of the valuation ring of $K$ over $Z\_p$?
There must be a straighforward reason why this is possible, but I can not figure it out...



Algebraic extensions of p-adic closed fields
Both the results you ask about, in the case of $p$-adic fields, follow from Krasner's lemma.
See for example, Proposition 3 and 4 of Lang, Algebraic Number Theory, p.43,44. 


The main point
is that two irreducible polynomials which are sufficiently close in the $p$-adic topology have roots which generate the same extension. 
The second clearly follows from this by approximating a polynomial with coefficients in (say) $\mathbb{Q}\_p$ by a polynomial with coefficients in $\mathbb{Q}$. To get the first result, one may assume the field is generated by a root of a monic irreducible polynomial with (say) $\mathbb{Z}\_p$ coefficients and then use the compactness of $\mathbb{Z}\_p$ (and approximation as above) to get finiteness.


