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.



uncountable algebraically closed field other than C ?
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. 


