Let $K/F$ be a Galois extension, and $f(x) ∈ F[x]$ be a nonzero polynomial whose complex roots are in $K$. Let $a_1,···,a_n$ be the distinct roots of $f(x)$. Let $ρ ∈ Gal(K/F)$. Prove that there is a bijection $roots(f(x)) → roots(f(x))$ given by $a_i → ρ(a_i)$.
Attempt: We need to show that if $a$ is a root of $f(x)$, then $ρ(a)$ is a root of $f(x)$. There was a theorem in my textbook that which says that if $g: K → \mathbb{C}$ is an embedding that fixes $F$, and $a ∈ K$ is algebraic over $F$ then $g(a)$ is a conjugate of $a$ over $F$. Is that useful here?
Hint: you don't need to use that theorem: instead, prove that $\rho(f(x)) = f(\rho(x))$ for any $x \in K$ (using the fact that $\rho \in \mathit{Gal}(K/F)$, so that $\rho(c_i) = c_i$ for each of the coefficients of $f(x) = c_0 + c_1x + \ldots c_mx^m \in F[x]$).