Question regarding a cyclic field extension over a field of positive characteristic.

Question

This is problem 6, b) in p. 297 of Hungerford's algebra. Suppose $F/K$ is a cyclic field extension of degree $p^n$, with $\text{char}(K) = p > 0$, it must be proven that there exists a cyclic extension $E/K$ of degree $p^{n+1}$. However, my doubt is in something stated in the hint of the problem. As $F/K$ is cyclic, there exists $\alpha \in F$ such that $$\text{Tr}(\alpha ) = 1$$ where $\text{Tr}$ is the trace operator of $F$ over $K$. It is posible to prove that $$\text{Tr}(\alpha^p - \alpha) = 0$$ which implies that there exists $\beta \in F$ such that $$ \beta - \sigma \beta = \alpha^p - \alpha,$$ where $\text{Gal} (F/K) = \langle \sigma \rangle$. The book states that $x^p - x - \beta$ must be irreducible over $F[x]$, which can be proven if it does not split in $F[x]$ by a theorem in the section of the problem, which states that $\forall \gamma \in F$, $x^p -x - \gamma$ either splits in $F[x]$ or is irreducible over $F[x]$. However, I am not able to see why $x^p - x - \beta$ does not split in $F[x]$.