Is $\mathbb R$ with usual euclidean topology, homeomorphic with some topological field of positive characteristic?

Question

Does there exists a topological field of positive characteristic which is homeomorphic with $\mathbb R$ with the usual topology ?

By homeomorphism here, I mean just topological homeomorphism, not necessarily preserving any algebraic structure.

Answer 1

Consider the map $\psi:x\mapsto x+1$ on a topological field $K$. It's a homeomorphism from $K$ to itself, and it has no fixed points. If $K$ has characteristic $p$ then $\psi$'s $p$-th power is the identity.

But on $\Bbb R$ every homeomorphism $\psi:\Bbb R\to \Bbb R$ without fixed points is strictly increasing and either $\psi(x)-x$ is always positive or $\psi(x)-x$ is always negative. So either $0<\psi(0)<\psi^2(0)<\psi^3(0)< \cdots$ or $0>\psi(0)>\psi^2(0)>\psi^3(0)> \cdots$. Either way, $\psi^p$ cannot be the identity.