Are $\Bbb R$ and $\Bbb C$ the only connected, locally compact fields?

Solution 1:

The statement seems to be a corollary of Theorem 21 from Lev Pontrjagin book “Continuous groups”. Unfortunately, I have only Russian version of it. But this book is classical, so it should have an English translation.

Solution 2:

We have to be careful, since fields with discrete topology are also locally compact! Conceptually, the proof that the only nondiscrete locally compact fields are of the type mentioned in the comments is not difficult, as I recall (but my recollection is not reliable). Use a Haar measure on the locally compact multiplicative group of the field. This measure is unique up to a multiplicative constant, let's call it $\mu$. Now take a compact set with nonzero measure, call it $X$. Let $\alpha$ be a nonzero element of the field, and compare $\mu(X)$ to $\mu'(X)=\mu(\alpha X)$. Here, $\mu$ and $\mu'$ both are Haar measures, and so must be related by a positive real constant, which we can call $|\alpha|$. This association, $\alpha\mapsto|\alpha|$, doesn't depend on the choice of $X$ nor on the original choice of $\mu$, and you have to verify that it is an absolute value (or almost), and now the remaining details escape me. But Weil starts his book “Basic Number Theory” with this topic, you can find it there.