I want to show that if an element in $\beta \in K$ where $|K|=p^n$ is a generator for $K^*$, i.e. has order $p^n-1$, then there is a generator $\alpha\in L$, $[L,K]=2$ i.e. of order $(p^n)^2-1$, such that $\alpha^{p^n+1}=\beta$.
I've tried rephrasing the question to understand it in a different way, but it hasn't helped. Essentially the question is the same as: Show that there is a root to $x^{q+1}-\beta$ in $L$ such that it is a generator of $L^*$ given that $\beta$ is a generator of $K^*$.
Initial thoughts: In a previous question, I now understand that every $\beta\in K$ can be written as $\alpha^{q+1}$ for some $\alpha\in L$ but it is not obvious that if $\beta$ is a $K^*$ generator, then the element $\alpha\in L$ that satisfies $\alpha^{q+1}=\beta$ necessarily is a generator for $L^*$. The converse is obvious, if $\alpha$ is a generator of $L^*$ then $\alpha^{q+1}$ is clearly in $K$ and is a generator because the lowest power that takes $\alpha$ to the identity is $q^2-1$ so we have to raise $\alpha^{q+1}$ to the power $q-1$ to do that, which means it is a generator for $K^*$. But the direction I want has been more difficult for me, because for a given generator of $K^*$, say $\beta$, then an element $\alpha\in L$ of order $q-1$ could work, because we obviously have $\alpha^{q+1}\in K^*$ and the lowest common multiple of $(q+1,q-1)=(q^2-1)/\delta$ where $\delta=1$ if $q=2^n$ and $\delta=2$ otherwise. Not sure where to go from here.