If $(f,g)=1$ , then prove for any $m,n\in \mathbb{Z}^+$, $(f^m,g^n)=1$ . $(f,g)=\gcd(f,g)$.
My try: Split $f$ to product of some irreducible polynomial. Like $f=ap_1^{\alpha_1}\cdots p_t^{\alpha_t} $. So do $g$. Then their irreducible polynomials won't have two same. So we can conclude $(f^m,g^n)=1$.
But it seems a little bit verbose for this simple question . Any more concise ways?
Suppose that there is an irreducible polynomial $p$ such that $p\mid f^m$ and $p\mid g^n$. Then, since $p$ is irreducible, $p\mid f^m\implies p\mid f$ and, for the same reason $p\mid g$. But that's impossible, since we're assuming that $\gcd(f,g)=1$. Therefore, $\gcd(f^m,g^n)=1$.