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.

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    $\begingroup$ $\mathbb{Q}_p$ is not $C_1$ (that's easy). It's not even $C_2$. Artin conjectured that it was but it was later disproved. Greenberg "Lectures on forms in many variables" is a good reference for your question. $\endgroup$ – Felipe Voloch Sep 17 '15 at 21:52
  • $\begingroup$ @FelipeVoloch Thank you very much for the comment. I was wondering if you have any comment in the general case (the first two questions). $\endgroup$ – user43198 Sep 18 '15 at 14:17

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 $C_1$.

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    $\begingroup$ What is $n$ here? $\endgroup$ – abx Sep 19 '15 at 4:57
  • $\begingroup$ @abx The degree of some algebraic extension of the residue field. $\endgroup$ – Felipe Voloch Sep 19 '15 at 6:55

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