I have the following question on my problem set.
Suppose that $\textsf{k}$ is a field and $\phi:\textsf{k}\to\textsf{k}$ is a homomorphism. Check that $\textsf{k}^\phi=\{x\in\textsf{k}:\phi(x)=x\}$ is a subfield of $\textsf{k}$. If $\operatorname{char}(\textsf{k})=p>0$ show that $\phi(x)=x^p$ is a homomorphism from $\textsf{k}$ to itself. Deduce that the subset $\textsf{k}^{\phi^n}=\{x\in\textsf{k}:\phi^n(x)=x\}$ is a subfield of $\textsf{k}$ with $p^d$ elements for some $d\leq n$. ($\phi^n=\phi\circ\phi\circ\cdots\circ\phi$ is $\phi$ composed with itself $n$ times.)
I have everything up the last deduction. I know that $\textsf{k}^{\phi^n}$ is a subfield and that if it is finite, then the number of elements it has is a power of $p$, but I can't see a why it is finite or why if $\textsf{k}^{\phi^n}$ does have $p^k$ elements, why $k\leq n$.
I'm not really sure where to go from here.
Could I use $\textsf{k}[t]$ to represent the $\textsf{k}^{\phi^n}$ as the zeros of, say, $g_n=\phi^n(t)-t=t^{p^n}-t$?
Could it be fruitful to think of the field extension $\textsf{k}/\textsf{k}^{\phi^n}$?