Question: On the proving following theorem about finite fields

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> $\mathbb{F}\_{p^n} \supset \mathbb{F}\_{p^d}$ iff $d|n$
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I have some problems. I know $\Rightarrow$ is trivial by considering extension degree.

But I have a problem on $\Leftarrow$ direction.

The textbook says

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> Suppose $d|n$ then if $x^{p^d} =x$ then also $x^{p^n} =x$, thus $\mathbb{F}\_{p^d} \subset \mathbb{F}\_{p^n}$.
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But i am not sure why $x^{p^d} = x$ iplies $x^{p^n} =x$.

For example $d|n$ implies $n=dk$ for some $k$, then $x^{p^d} = x^{p^{nk}} = x$ then why this implies $x^{p^n}=x$?

Answer: 
We have $$x^{p^{2d}}=(x^{p^d})^{p^d}=(x)^{p^d}=x.$$ Now you can use induction to get your claim.
Induction using $$ x^{p^{kd}}=x^{p^dp^{(k-1)d}}=(x^{p^{d}})^{p^{(k-1)d}}=x^{p^{(k-1)d}}$$