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
?