Valuation rings of $k(x)$ when $k$ is algebraically closed

Question

Let $k$ be an algebraically closed field. I am asked to find all valuation rings of $F=k(X)$.
I know what are all the valuation rings of $F$ containing $k$. But my question is whether every valuation ring of $F$ will contain $k$?
An extension of the question ìs that suppose $(R,m)\subseteq k(X)$ is a local subring of $k(X)$. Does that imply $R$ must contain $k$? Is it true if $k$ is algebraically closed.
The problem is taken from Daniel Bump's Algebraic Geometry. Any help will be appreciated.

Answer 1

No, any non-trivial valuation on $k$ will extend to a valuation on $k(X)$, and algebraically closed fields typically have lots of non-trivial valuations.

For instance, $\mathbb{C}$ is (non-canonically) isomorphic as a field to any $\mathbb{C}_p$, and therefore has a $p$-adic valuation on it, which will then extend to a valuation on $\mathbb{C}(X)$ which is non-trivial on $\mathbb{Q}$ (let alone $\mathbb{C}$).