How to Prove that $\operatorname{Aut}(\mathbb C/\mathbb Q)$ is infinite?
I had already proved that $\operatorname{Aut}(\mathbb R/\mathbb Q)$ is trivial group containing identity use continuity argument.
But When I thought about $\operatorname{Aut}(\mathbb C/\mathbb Q)$ problem I could not even start.
I had just started course in field theory in which we have done Galois group definition and some example.
Please give me hint so that I can solve the above problem.
Fixing a transcendency basis $I$, we have $|I|=\mathfrak{c}=2^{\aleph_0}$. Then any permutation of $I$ extends to a (non-unique) automorphism. Hence the cardinal of this automorphism group is non only infinite, but equal to $2^{\mathfrak{c}}=2^{2^{\aleph_0}}$.
More generally, for any algebraically closed field $K$ of uncountable cardinal $\alpha$, the cardinal of $\mathrm{Aut}(K)$ is $2^\alpha$. (This is also true whenever $\alpha$ is countable, although the previous argument only applies when the transcendence degree is infinite.)