Let $\alpha$ be an algebraic element over a field $K$. If $f(x) = x^n -a, g(x)=x^m -a$ are irreducible polynomials in $K[x]$ such that $f(\alpha^m) = g(\alpha^n) =0$ and $n,m$ are coprime; I'm asked to find the degree of the extension $K(\alpha) : K$ and the minimal polynomial of $\alpha$ in $K$.
I don't really know how to work on the problem since, from my point of view, the solution itself should depend on the field $K$ and the element $a\in K$ (unless the condition of irreducibility for both $f,g$ manages these scenarios). I would really appreciate any suggestion.
Note that $K(\alpha^m, \alpha^n)$ contains both $K(\alpha^m)$ and $K(\alpha^n)$, so $[K(\alpha^m,\alpha^n):K]\ge mn$ because $\gcd(m,n)=1$. But $[K(\alpha^m,\alpha^n):K]$ is also bounded above by $[K(\alpha^m):K][K(\alpha^n):K]=mn$, so $[K(\alpha^m,\alpha^n):K]=mn$.
Now, since $\gcd(m,n)=1$, there are $p,q\in\mathbb{Z}$ such that $pm+qn=1$. Then $\alpha=(\alpha^m)^p (\alpha^n)^q\in K(\alpha^m,\alpha^n)$. This shows that $K(\alpha)=K(\alpha^m,\alpha^n)$, so $[K(\alpha):K]=mn$. The minimal polynomial of $\alpha$ over $K$ is then $x^{mn}-a$ because it is a polynomial of degree $[K(\alpha):K]$ that vanishes $\alpha$.