Suppose that $K$ is a field of characteristic $p\gt 0$, where $p$ is a prime, and suppose that the Frobenius homomorphism $\phi: K\to K$, where $\phi(\alpha)={\alpha}^p$, $\forall \alpha\in K$ is surjective. Show that every non-constant polynomial $f(x)\in K[x]$ is separable.
I have difficulties to find the connection between the given and the conclusion.
All I can see is that,
$1.$ Since $\phi$ is a field homomorphism, it is injective automatically. So $\phi$ is actually a bijection.
$2.$ If $\operatorname{char}(K)=p$, then all polynomials in the form of $f(x)=x^p-k\in K[x]$ is irreducible.
Then I have no idea what to do. Could anyone help me here?
Hint: $k = a^p$ for some $a$, so $x^p - k = (x-a)^p$.