Is a normal irreducible separable polynomial of degree $p$ over a field of characteristic $p$ necessarily Artin-Schreier? If it is true, I am also wondering whether we can generalize this in the following sense.
Let $K$ be a perfect field of characteristic $p$. We say that a normal irreducible polynomial $f(X) \in K[X]$ is cyclic if the corresponding Galois group is cyclic. Let $f(X) \in K[X]$ be cyclic of degree $p^n$, $f(\theta) = 0$. Then $K(\theta)/K$ is a cyclic Galois extension of degree $p^n$, hence a tower of Artin-Schreier extensions $K \subsetneq K_1 \subsetneq \cdots \subsetneq K_{n-1} \subsetneq K_n = K(\theta)$. Do we always have that $\theta$ is an Artin-Schreier generator of $K_n$ over $K_{n-1}$?
I am not familiar with the Artin-Schreier-Witt theory, and I apologize if the question is a standard result. Please provide me with a reference if that is the case.