We know that $(\mathbb{Q},+,\times)$ is a subfield of $(\mathbb{R},+,\times)$. It is easy to see that the automorphism of $\mathbb{Q}$ is only the identity. For a quick proof, lets go through the main steps:
- $f(1)=1 \Longrightarrow f(n)=n$ for all $n \in \mathbb{N}$.
$f(-1)=-1$ which says that $f(x)=x$ for all $x \in \mathbb{Z}$.
$\displaystyle f \Bigl(\frac{p}{q}\Bigr) = \frac{p}{q}$, where $q \neq 0$.
One, then uses the continuity of $f$ and the denseness of $\mathbb{Q}$ to prove that the Automorphism of $\mathbb{R}$ is also trivial.
My Question Given a subfield $K$ of $\mathbb{C}$ is it and an automorphism of $K$ can it be extended to the whole of $\mathbb{C}$.