Automorphism in field

Question

I have a math problem I can't solve I need to show that if $K$ is a field and $x \in K$, $f(x) = x$ ($f$ is a function) only true for any automorphism in $K$ if $x$ is in the primefield of $K$. Can anyone explain this to me?

Answer 1

Let $K$ be the subfield of $\mathbb{R}$ given by $K=\mathbb{Q}(\theta)$, where $\theta$ is the unique real root of $p(t)=t^3+t+1$.

Then any automorphism of $K$ fixes $\theta$ (since the other two roots of $p$ are non-real), but $\theta\notin \mathbb{Q}$, which is the prime field of $K$.

In fact, for this field $K$, the identity automorphism is the only automorphism, hence any element of $K{\setminus}\mathbb{Q}$ would yield a counterexample to the proposed claim.