A field $\mathbb{K}$ is said to have the extension property if every automorphism of $\mathbb{K}(t)$ is an extension of an automorphism of $\mathbb{K}$, where $t$ is a variable. It is equivalent to the next property:
For every automorphism $\phi:\mathbb{K}(t)\rightarrow\mathbb{K}(t)$, $\phi(\mathbb{K})\subset\mathbb{K}$.
I think $\mathbb{C},\mathbb{R}, \mathbb{Q}$ and $\mathbb{Z}_{p}$ have the extension property. Do you know any field that does not have the extension property?