As I don't really know where else to publish this (mathoverflow etc do not seem to have a section for it), here comes a wrong construction of a field of characteristic 1. Still, it is wrong in a somewhat interesting way. Maybe as an exercise for an introductory lecture in mathematical logic or something.
So to make things easy, we work with ZFC. We could probably use weaker systems, but this would just make things harder to express.
Now, let be the first-order theory of Zermelo-Fraenkel set theory, and let us assume that . We define first-order constants , , , , with the axioms
Now define and , where the first addition is ordinal addition, and the second addition is addition in the field . Essentially, this defines that means adding for times. This is a simple recursive definition, so it is justified according to the recursion theorem. We add the axiom
let us call the resulting theory . Obviously, . Now define the axioms , and , and consider .
Every finite subset of is satisfiable, because there are fields of arbitrarily large characteristic. Hence, because of the compactness theorem, , and hence, there is a model with a constant such that
We can "lift" out of its model , by setting , similar for the other constants.
Now comes the exercise: Why is not a field of characteristic 1?
I could actually imagine that, though the actual characteristic of should be 0 (if I am correct), this thing has some properties that one wants from fields with characteristic 1, so it might be interesting to look at (but if it is, there certainly are people who already do this).