Can we start from the restriction, and continue with the normal function?

Question

Suppose that we have the field automorphism $\phi :C\rightarrow C$ with $\phi (a)=b$.

When we consider the restriction $\phi|_K :K\rightarrow C$, and we have that $a^n\in K$, do we have the following? $$\phi|_K (a^n)=\phi|_K (a\cdot a \cdot \ldots \cdot a)=\phi (a)\cdot \phi (a)\cdot \ldots \cdot \phi (a)=b\cdot b\cdot \ldots \cdot b=b^n\in K$$

Or can we not start from the restriction at K, and continue with the normal function $\phi$ ?

Answer 1

All of the equalities that you wrote are true except the last one. We do not necessarily have $b^n\in K$.