Can we make extension fields with non-finite numbers? E.g. does something like $\mathbb R (\aleph)$ make sense?

Question

Cardinals have a well-defined arithmetic, but I don't have enough background knowledge to check if this notion makes sense.

Answer 1

I think you can follow the usual ideas, but if it were possible to have an extension field, say $\mathbb{R}(\kappa),$ then all such fields would be isomorphic for all infinite cardinals $\kappa$, since these are all idempotent ($\kappa^n=\kappa$ for all finite $n$). Anyway, although my intuition fears the answer might be not, I still didn't think of a reason that would impede that $\mathbb{R}^2$ with sum $(a,b)+(c,d)=(a+c,b+d)$ and product $$(a,b)\cdot(c,d)=(ac,ad+bc+bd)$$ (such would result its product, I believe) be a field. (It's not immediate for me that a multiplicative inverse will nor will not exist).

Answer 2

I'm going to say no. $\newcommand{\RR}{\mathbb{R}}\RR(\alpha)$ has two possible meanings that I can think of depending on context. If $\alpha$ is an element of an (already existing) extension field $K$ of $\RR$, then $\RR(\alpha)$ is the smallest subfield of $K$ containing both $\RR$ and $\alpha$. If $\alpha$ is an indeterminate, then $\RR(\alpha)$ is the field of rational functions in $\alpha$.

$\aleph$ isn't either of those things, nor can it be. $\aleph^2=\aleph$ according to the usual rules of cardinal arithmetic, which would imply that $\aleph=0$ or $\aleph=1$ if we were to regard this as taking place in a field. So it is essentially impossible.