I was trying to prove the following theorem:
A field $K$ of order $q=p^r$ contains a subfield $K'$ of order $q'=p^k$ if and only if $k\mid r$.
My attempt at the proof was:
$K'\setminus\{0\}$ is a subgroup of $K\setminus\{0\}$. So, $p^k-1\mid p^r-1$.
From the exponent gcd lemma [$\gcd(a^x-1,a^y-1)=a^{\gcd(x,y)}-1$ when $a,x,y\in\mathbb N,a\geq 2$.]
We have that $k\mid r$.
Is this proof correct? In fact, I'm worried about this particular statement:
$K'\setminus\{0\}$ is a subgroup of $K\setminus\{0\}$.