Let denote the first infinite cardinal – that is, the set of non-negative integers. Let be the smallest prime number, and let enumerate all prime numbers in ascending order.
Let be a free ultrafilter on . We consider the field
This uses the standard notation for considering the equivalence relation on where we have for if and only if . (It is easy to verify that this is an equivalence relation.) On we use component-wise addition and multiplication. It is a standard exercise to show that whenever and then , and the same holds for multiplication. So the operations are well-defined on the quotient , and it is another standard exercise to show that is indeed a field (as opposed to , which has lots of zero divisors). Moreover, is uncountable and has characteristic 0.
I don’t know if and where this field has been studied, or if there is a well-known field that is isomorphic to .
There are several questions I cannot answer, and I would be grateful for any hints on them.
- If we take different free ultrafilters , can it happen that is not isomorphic to ?
- If yes: Suppose are free ultrafilters such that and are isomorphic fields. What can be said about ? For instance, do they have to be in relation with respect to the Rudin-Keisler ordering?
- Can be made into an ordered field?
- Are there surjective group homomorphisms from the additive group of to the additive group of , or vice versa? What about the multiplicative group of ?