$\mathbb{F}_{p^d}\subseteq\mathbb{F}_{p^n}$ if and only if $d$ divides $n$

Question

I am trying to solve the following exercise of Dummit and Foote Book(page # 551).

Let $a>1$ be an integer. Prove for any positive integers $n,d$ that $d$ divides $n$ if and only if $a^d-1$ divides $a^n-1$. Conclude in particular that $\mathbb{F}_{p^d}\subseteq\mathbb{F}_{p^n}$ if and only if $d$ divides $n$.

I did the first part and I know that for all $\alpha\in \mathbb{F}_{p^d}$, $\alpha^{p^d}=\alpha$. How can I apply the first part for the second? Any help is greatly appreciated. Thank you.

Answer 1

$\mathbb F_{p^n}$ is the splitting field of $x^{p^n}-x$.

Answer 2

If $\mathbb{F}_{p^d}\subseteq\mathbb{F}_{p^n}$, then $n=[\mathbb{F}_{p^n}:\mathbb{F_p}]=[\mathbb{F}_{p^n}:\mathbb{F}_{p^d}][\mathbb{F}_{p^d}:\mathbb{F_p}]=[\mathbb{F}_{p^n}:\mathbb{F}_{p^d}]d$, and so $d$ divides $n$.

If $d$ divides $n$, then $\mathbb{F}_{p^d}^\times$ is a subgroup of $\mathbb{F}_{p^n}^\times$ and so $\mathbb{F}_{p^d}\subseteq\mathbb{F}_{p^n}$. Here we use that $\mathbb{F}_{p^n}^\times$ is cyclic and that there is at most one finite field of a given size.