I am working on a model theory problem that has very little to do with field theory.
I basically want to know if and why every algebraically closed field of characteristic 0 is uncountable. I cannot think of a reason why this is true, but given the problem I am working on, I suspect it must be.
I know many abstract algebra theorems assume uncountability. I also know that the cardinality of the algebraic closure of a field F is $max\{\aleph_0, |F|\}$.
Is it true that algebraically closed fields of characteristic 0 are uncountable?
Best Answer
$\mathbb Q$ has characteristic $0$ and is countable by a famous spiral argument. As you correctly state, the cardinality of the algebraic closure of a field $F$ is $\max\{\aleph_0, |F|\}$, so the cardinality of the algebraic closure of $\mathbb Q$ is $\aleph_0$.