Help with problem of the book Algebra of T. Hungerford

Question

This is a problem in “Algebra” by T. Hungerford:

7. If $|K|=q$ and $f\in K[x]$ is irreducible, then $f$ divides $x^{q^n}-x$ if and only if $\deg f$ divides $n$.

I found it difficult to solve this exercise. I do not know if maybe they already have been solved in this forum. please any help will be appreciated .

Answer 1

Hints, assuming $\;q=p^r\;,\;\;p\;$ a prime:

== The field with $\;q^n\;$ elements is exactly the set of all the roots of $\;x^{q^n}-x\in\Bbb F_p[x]\;$ in some algebraic closure of $\;\Bbb F_p\;$

== The field $\;\Bbb F_{p^m}\;$ is a subfield of $\;\Bbb F_{p^n}\;$ iff $\;m\mid n\;$

You may want now to mull also on the hint given to you by user26857