I have to demonstrate that there is a infinite number of automorphisms of $\mathbb C$, using Zorn's lemma; I found some material online but it didn't match well with the program of my course. Following the suggestions of the professor, I have to show that for every subfield $F$ of $\mathbb C$ and for every field automorphism $\phi$ of $F$, there is an extension of $\phi$ to an automorphism of $\mathbb C$.
Let's consider the set $A$ of automorphisms $\eta : E \to E$, with $F\le E$ and $\eta |_F = \phi$: we can estabilish a partial order relation in $A$ putting $\alpha \le \beta $ if $K\le L$, where $\alpha : K \to K$, $\beta : L \to L$, with $F\le K,L$ and $\alpha |_F = \beta |_F =\phi$. Now I should show that any totally ordered subset of $A$ has an upper bound in $A$, in order to prove that there is a maximal element in $A$. However I have no ideas, since the only condition on $F$ is that $\mathbb Q \le F$, and in general a succession $\mathbb Q \le F_1 \le F_2 \dots$ of subfields of $\mathbb C$ is not finite. Can you give me a hint on what I am missing? Thanks
For your order relation it is better to write $\alpha\leq\beta$ if and only if $K\subset L$ and $\beta_{\mid K}=\alpha$, if $(L_\alpha)$ is totally ordered, the maximal element is $\cup_\alpha L_\alpha$.