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).

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...