Algebraically closed field of cardinality greater than $\mathfrak c$

Question

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?

Answer 1

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, when this phenomenon arose a few weeks ago at our seminar.

Answer 2

As a purely algebraic alternative to model theoretic approaches to nonstandard algebraic closure in characteristic $0$, we can also constructively build them in the regular old universe out of the ordinals.

Let $\alpha>0$ be an ordinal, and consider the field of fractions $Frac(\mathfrak{G}(\omega^{\omega^\alpha}))=\mathbb{Q}^{\omega^{\omega^\alpha}}$ of the Grothendieck ring $\mathfrak{G}(\omega^{\omega^\alpha})=\mathbb{Z}^{\omega^{\omega^\alpha}}$ of the $\delta$-number $\omega^{\omega^\alpha}$ under natural (Hessenberg) operations. This is a non Archimedean ordered field (thusly it properly contains the rationals) with no nontrivial roots where all the ordinals up to $\omega^{\omega^\alpha}$ are now treated as the 'natural numbers' and $\mathbb{Z}^{\omega^{\omega^\alpha}}$ is treated as the 'integer subring' of our field.

We can then move into an algebraic closure $\overline{\mathbb{Q}}^{\omega^{\omega^\alpha}}$, where we allow 'polynomials' to really be elements of the monoid algebra $\mathbb{Q}^{\omega^{\omega^\omega}}(X^{\omega^{\omega^\alpha}})$ -- these are expressions of the form $$\sum_{i<n}q_iX^{\alpha_i}$$ with $q_i\in\mathbb{Q}^{\omega^{\omega^\omega}}$ and $\alpha_i<\omega^{\omega^\alpha}$ for all $i<n$. This gives us '$\beta^{th}$ roots' for members of $\mathbb{Q}^{\omega^{\omega^\alpha}}$ for all $\beta<\omega^{\omega^\alpha}$, as opposed to only finite roots which are provided by algebraic closure using standard polynomials.

There is some subtlety here when trying to approach this without model theory guaranteeing the existence of a way to evaluate $q_iX^{\alpha_i}$ as a function on $\mathbb{Q}^{\omega^{\omega^\alpha}}$ when $\omega\leq\alpha_i$ -- we can get around this by embedding $\mathbb{Q}^{\omega^{\omega^\alpha}}$ in the Surreals $N_0$ and looking at what the Surreal exponential function does to its image.**

You can also do modular arithmetic in $\omega^{\omega^\alpha}$ for ordinals $\beta\geq\omega$ and proceed to produce a nonstandard $\mathbb{Q}^{\omega^{\omega^\alpha}}_p$ for $p\geq\omega$ prime, and then algebraically close that too. We could also just use all of the ordinals instead of stopping at $\omega^{\omega^\alpha}$, in which case we produce a proper class sized version of the above.

I'm currently researching these notions to better understand the Surreals, but they seem to provide a constructive model of many nonstandard constructions usually accessed through model theoretic techniques.

*This used to claim that $\mathbb{Q}^{\omega^{\omega^\alpha}}$ was essentially a nonstandard model of the rationals, which is incorrect (thanks to nombre on this).

** I am currently checking if $\mathbb{Q}^{\omega^{\omega^\alpha}}$ embedded in $N_0$ 'preserves exponentiation' to well-define it on the preimage; I believe it should, but this is just intuition for now.

Answer 3

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.