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?



Algebraically closed field of cardinality greater than $\mathfrak c$
Here is a different way to think about how these fields arise.


In characteristic zero, all such fields in any uncountable cardinality $\kappa$ can be viewed as arising as a nonstandard version of the algebraic closure the rationals. 


That is, take any nonstandard model of arithmetic $M$ of uncountable size $\kappa$, or an $\omega$-nonstandard model of set theory with $\kappa$ many natural numbers, if you like. Let $\mathbb{Q}^\*$ be the nonstandard version of the rational field as $M$ sees it, and let $\overline{\mathbb{Q}^\*}$ be the algebraic closure of $\mathbb{Q}^\*$ as it is constructed in $M$. This is an algebraically closed field of characteristic zero and of the desired cardinality $\kappa$. 


In particular, even the complex field $\mathbb{C}$ itself can be seen as the algebraic closure of the rationals $\mathbb{Q}^\*$ taken inside a nonstandard model of arithmetic or set theory, and $\mathbb{Q}^\*$ is the quotient field there of its integer part $\mathbb{Z}^\*$. 


In this sense, every uncountable algebraically closed field of characteristic zero is exactly a nonstandard version of the algebraic closure of $\mathbb{Q}$. 


Note that the algebraic closure of a field as computed inside $M$ is more generous than the actual algebraic closure, since a nonstandard model will be wanting to include all the roots of the various nonstandard polynomials, as well as the standard ones.


I made a blog post, [Nonstandard models of arithmetic arise in the complex numbers](http://jdh.hamkins.org/nonstandard-models-of-arithmetic-arise-in-the-complex-numbers/), when this phenomenon arose a few weeks ago at our seminar.


