By Lang's theorem, a complete valued field which is the fraction field of a discrete valuation ring with an algebraically closed residue field is quasi-algebraically closed (or $C\_1$). 


How much is known about the converse?
Is there a criterion/almost exhaustive list of complete valued fields, which is quasi-algebraically closed? For example, is $\mathbb{Q}\_p$ quasi-algebraically closed?


Any reference dealing with such examples will be most welcome.



On quasi-algebraically closed fields
If you have a dvr whose residue field is not algebraically closed, then there is a norm form in the residue field, so a form of degree $n$ in $n$ variables with no non trivial zero. Lift this form to a form $f$ over the ring. Now let $\pi$ be an element of value one and consider the form $f(x\_1,...,x\_n)+\pi x\_{n+1}^n$, a form of degree $n$ in $n+1$ variables with no-notrivial zero in the ring (hence in the fraction field). So the field is not $C\_1$.


