I'm trying to solve the following problem from Algebra by Siegfried Bosch (english version below):
Es seien $m,n$ teilerfremde positive ganze Zahlen. Ist dann $L/K$ eine Körpererweiterung vom Grad $m$, so hat jedes Element $a\in K$, welches eine $n$-te Wurzel in $L$ besitzt, bereits eine $n$-te Wurzel in $K$.
My attempt at a translation:
Let $m,n$ be positive integers with $\mathrm{gcd}(m,n) = 1$. If $L/K$ is a field extension of degree $m$, then show that for any element $a\in K$ having an $n$-th root in $L$, there already is an $n$-th root of $a$ in $K$.
Since it's in the chapter, where trace and norm are defined, I'm guessing the norm will have to be used. So the question is: how?
The only (obvious) thing I can see:
If $b \in L$ is an $n$-th root of $a \in K$, then we have that $a^m = N_{L/K}(a) = N_{L/K}(b^n) = (N_{L/K}(b))^n$ and $N_{L/K}(b)\in K$. But I don't know whether (or how) this helps.
I don't seem to be able to come up with any good ideas, so I would appreciate some pointers. But please don't write down a full solution, if possible. I really would like to learn something here, so if you could give me just a hint, that would be great!
Thanks a lot.