TITLE: Is there a measurable isomorphism ${\mathbb C}\to {\mathbb C}_p$?
QUESTION [5 upvotes]: Let $p$ be a prime and ${\mathbb C}_p$ be the completion of the algebraic closure $\overline{{\mathbb Q}_p}$. This field is isomorphic to $\mathbb C$. Both fields come with natural absolute values but have very different topologies. There are no continuous field isomorphisms, but that does not exclude  Borel-measurable ones (measurable in both directions, that is). Do those exist?
A related question is the question for measurable automorphisms of $\mathbb C$. Are there more than the complex conjugation?

REPLY [9 votes]: $\mathbb C$ and $\mathbb C_p$ are separable complete metric spaces (Polish spaces).  They are Polish groups under addition.  A Borel-measurable homomorphism between Polish groups is continuous.  (Sometimes called "automatic continuity".) So any such Borel isomorphism must be a homeomorphism.  But $\mathbb C$ is connected while $\mathbb C_p$ is totally disconnected, so they are not homeomorphic.  We did not require field isomorphism, only group isomorphism for addition.