Fields on uncountable sets

Question

I recently found out that when the underlying set is countably infinite, the only field structure that can be added to it will make it isomorphic to $\mathbb{Q}$ (is this correct?)
So, i was wondering, how many (up to isomorphism) different structures can be given to a set of cardinality same as $\mathbb{R}$ (of the continuum)? I only know of $\mathbb{R}$, $\mathbb{C}$ and the fields $\mathbb{Q}_p$ of p-adic numbers, for each prime p. Are there more?

Answer 1

User Crostul mentions an important class of counter-examples to your first question, and also the limit of finite fields can be countable with characteristic $p$.

Consider the characteristic $0$ case. A countably generated transcendental extension of $\mathbb{Q}$ will be countable, and assuming the continuum hypothesis it can be at most generated by $2^{\aleph}$ elements. This leaves the case that $K > \mathbb{Q}$ is the function field of some quotient of $\mathbb{Q}[x_i]$ where $i$ ranges a set with cardinality of the continuum. By similar arguments, the case of characteristic $p$ can be given a similar description, which doesn't really shed light onto the structure of these fields. Still, it makes it clear (I think) that many such fields exist.