Is there any "wellknown" 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.
Is there any "wellknown" 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. 


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. 


The algebraic closure of the padic field $Q_p$ is also of interest. One may even want to consider the completion (with respect to the padic 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 padic analog of $C$. 

