# On quasi-algebraically closed fields

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.

• $\mathbb{Q}_p$ 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. – Felipe Voloch Sep 17 '15 at 21:52
• @FelipeVoloch Thank you very much for the comment. I was wondering if you have any comment in the general case (the first two questions). – 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 $C_1$.
• What is $n$ here? – abx Sep 19 '15 at 4:57