TITLE: uncountable algebraically closed field other than C ?
QUESTION [9 upvotes]: Is there any "well-known" algebraically closed field that is uncountable other than $\mathbb{C}$ ?
The algebraic closure of $\mathbb{C}(X)$ would work, but is it meaningful, is this field used in some topics ?
Have you other examples ?
Thank you.

REPLY [23 votes]: The algebraic closure of $\mathbb{F}_p((t))$ is uncountable of characteristic $p$.  It comes up naturally in number theory and algebraic geometry.
For every characteristic $p \geq 0$ and uncountable cardinal $\kappa$, there is up to isomorphism exactly one algebraically closed field of characteristic $p$ and cardinality $\kappa$.  The examples of $\mathbb{C}$ and closures of Laurent series fields as above give you the ones of continuum cardinality and all characteristics.  Indeed I do not know any specific reason to consider algebraically closed fields of larger than continuum cardinality.

REPLY [7 votes]: The algebraic closure of the p-adic field $Q_p$ is also of interest. One may even want to consider the completion (with respect to the p-adic absolute value) of this algebraic closure.
The resulting field is both complete and algebraically closed. It is denoted by $C_p$, and is considered as an p-adic analog of $C$.