abstract algebra - Characterising the irreducible polynomials in positive characteristic whose roots generate the (cyclic) group of units of the splitting field

For a nonzero element $\alpha \in \mathbb F_{p^n}$ (the finite field of cardinality $p^n$) is there a simple criterion to tell whether $\alpha$ is a generator of the cyclic group $\mathbb F_{p^n}^\times$ by looking at the minimal polynomial of $\alpha$ over $\mathbb F_p$?

Here is what I know so far:
Let $n\ge 1$ be an integer, $p$ be a prime and $\mathbb F_p = \mathbb Z/p\mathbb Z$ be the prime field of characteristic $p$. It is known that there is a unique (upto isomorphism) field $\mathbb F_{p^n}$ of cardinality $p^n$ which is the splitting field of $g (x) = x^{p^n} - x$.

If $f (x) \in \mathbb F_p [x]$ is monic irreducible of degree $n$ then $f (x)$ is separable and $\mathbb F_{p^n} \cong \mathbb F_p [t]/(f (t))$ is also the splitting field of $f (x)$.
Further if $\alpha$ is any root of $f (x)$ in $\mathbb F_{p^n}$ then

• $\mathbb F_{p^n} = \mathbb F_p (\alpha)$ (so $\alpha$ is a primitive element of the field extension $\mathbb F_{p^n}/\mathbb F_p$)


• $m_{\alpha, \mathbb F_p} (x) = f (x)$ (the minimal polynomial of $\alpha$ over $\mathbb F_p$)


• $f (x) = \prod\limits_{i = 0}^{n - 1} (x - \alpha^{p^i})$

Also, there are exactly

• $M (n, p)/n$ distinct degree $n$ monic irreducible polynomials in $\mathbb F_p [x]$


• $M (n, p) = \sum\limits_{d \mid n} p^{n/d} \mu (d)$ distinct primitive elements $\beta$ of $\mathbb F_{p^n}/\mathbb F_p$


• $p^n - 1$ elements in the cyclic group $\mathbb F_{p^n}^\times$


• $\phi (p^n - 1)$ generators of the cyclic group $\mathbb F_{p^n}^\times$

It is clear that each generator $\beta$ of the cyclic group $\mathbb F_{p^n}^\times$ is an element of degree $n$ over $\mathbb F_{p}$, i.e. $\deg m_{\beta, \mathbb F_p} (x) = n$, and hence is a primitive element of the field extension. It is also clear that all the other roots $\beta^{p^i}$ of the minimal polynomial $m_{\beta, \mathbb F_p} (x)$ are also generators of the cyclic group since $\gcd (p^i, p^n - 1) = 1$.

However in general the number of cyclic generators $\phi (p^n - 1)$ is much less than the number of primitive elements of the extension $M (n, p)$ (for example, for $n = 2$, $\phi (p^2 - 1)$ will be much less than $M (2, p) = p^2 - p$.

So now the question is:

1. Characterise all the monic irreducible polynomials $f (x) \in \mathbb F_p [x]$ whose roots are cyclic generators of $\mathbb F_{p^n}^\times$.


2. If $\beta$ is a cyclic generator of $\mathbb F_{p^n}^\times$ what are the indices $0 \le k < p^n$ for which the degree of $\beta^k$ over $\mathbb F_p$ is $n$ (or in general any $d \mid n$)?

P.S. I don't know any Galois theory, so if you use any of that, then I would appreciate if a reference is given.