Field with $\aleph_k$ elements

Question

For simplicity (at least I think), first assume the GCH. In algebra, we have the following:

• $\mathbb{R}$, $\mathbb{C}$ and a like have a cardinality of $\aleph_1$.
• $*\mathbb{R}$, $*\mathbb{C}$ and a like (hyperreals, surcomplex) have a cardinality of ${\aleph_2}$ (they can constructed as sequences modulo some ultrafilter, to say it crudely).

Question is: Can you make a set of cardinality $\aleph_k$ ($k>2$) into a field and what is known about them ?

Other question of course is (I think a little bit harder): What is known when we drop GCH is this story ?

Answer 1

For a cardinal number $\kappa\geq {\aleph_0}$, the quotient field of the polynomial ring ${\mathbb F}_2[X_\lambda\ |\ \lambda<\kappa]$ has cardinality $\kappa$.

Answer 2

It is incorrect that the hyperreals have cardinality $\aleph_2$. In fact, it is provable in $\sf ZFC$ that the hyperreals have the same cardinality as the reals (which may or may not be $\aleph_1$ depending on $\sf GCH$ assumptions).

To get larger fields, you can use the Lowenheim-Skolem theorem, which ensure that if there is one infinite model of a first-order theory, then there is an infinite model of every infinity cardinality.

More explicitly, we can take $\Bbb Q[X]$, where $X$ is a set of any cardinality, and quotient by a maximal ideal. This ensures that we get a field of cardinality $\max\{|X|,\aleph_0\}$.