Let $F$ be a field with an $n$-th root of unity. Let $a,b ∈ F$ such that $f(x) = x^n - a$ and $g(x) = x^n - b$ are irreducible. Show that $f$ and $g$ have the same splitting field, iff $b=c^na^r$ for some $c∈F$, $r∈\Bbb N$ and $gcd(r,n)=1$
What I do is assign $L_f$ and $L_g$ to the splitting fields of $f$ and $g$ respectively.
From what I managed to find online, it all involves Galois fields, which we haven't reached in our lecture and I doubt they would be included in the solution, if they are not yet covered.
For $\Leftarrow$ I guess, I can rewrite $g$ as $g=x^n-c^na^r$. Then $\sqrt[n]{c^na^r} \in L_g \Rightarrow c\sqrt[n]a^r \in L_g$. It also obviously holds that $c\sqrt[n]a^r \in L_f$ since $c \in F$ and $\sqrt[n]a \in L_f$ from $f(x) = x^n - a$
But for $\Rightarrow$ I have no idea.
Any help would be appreciated.