TITLE: Algebraically closed field of cardinality greater than $\mathfrak c$
QUESTION [10 upvotes]: It is a 20th century result that there exists only one algebraically closed field for a given characteristic $p$ and cardinality $\kappa>\aleph_0$, up to isomorphism. Is there a better way to imagine such fields, other than adjoining $\kappa$ transcendental elements to $\mathbb Q$ or $\mathbb F_p$ and taking the algebraic closure?
In characteristic 0, do these fields enjoy some nice topological properties? Can you maybe do analysis on them?
I have not found much about fields of cardinality greater than $\mathfrak c$, not even about the "next simplest case" $2^\mathfrak c$. Do you know of an references?

REPLY [2 votes]: In the caracteristic zero case, I suggest one way to look at this is to choose a prescribed real closed subfield $R_{\kappa}$ of said algebraically closed field $F_{\kappa}$ of cardinal $\kappa$ that one finds particularly interesting, so that geometric intuition (for instance basically that of a plane) or "analytic" intuition may apply to $F_{\kappa}$.
Of course each $R_{\kappa}$ appears "as an isomorphic copy" in each algebraically closed field of cardinal $\kappa$, but not all of them are isomorphic, so this is a means to distinguish ordered pairs $(F_{\kappa},R_{\kappa})$ between them. 
Each choice naturally transfers some structure from $R_{\kappa}$ to $F_{\kappa}=R_{\kappa}\oplus R_{\kappa}.\sqrt{-1}$, be it an absolute value $F_{\kappa}\rightarrow R_{\kappa}^{\geq 0}$ (and thus a nice sequential uniform structure), a valuation extending that of $R_{\kappa}$, a Hahn series structure, a transfer principle, order related saturation properties, an exponential map...
That said I don't think there is a distinguished real closed subfield of $F_{\kappa}$ in a field-theoretic standpoint in general.