Other ways to deduce Cyclicity of the Units of certain groups? The group of units of the rings $\text{GF}(p^r)$ and $\mathbb{Z}/p^r\mathbb{Z}$ are both cyclic (except for the exception of prime powers are not cyclic when $p=2$ and $r\ge 3$). This is a strong result which I have been using a lot but I don't understand it properly.
I would like to know more alternative proofs of these results. If there are any generalizations (e.g. can both cases be treated together?) and specific cases (e.g. is there something special about the intersection $\text{GF}(p) = \mathbb{Z}/p\mathbb{Z}$.
I hope that question doesn't seem vague, it's more like a couple of questions together - I will greatly appreciate anything on this topic so thanks very much!

Update
Inspired by lhf's answer to a question about Wilson's theorem is there any way to prove that if the product of all elements of a group are $-1$ then the units group is cyclic?

It seems like you aren't aware of the result yoyo cites in the comments, so I think it's worth pointing out as a start.
Theorem: A finite subgroup of the multiplicative group of any field is cyclic.
Proof. Let $G$ be a finite subgroup of $F^{\ast}$, $F$ a field, and $n = |G|$. Then the elements of $G$ must be precisely the roots of the polynomial $x^n - 1$. We have the factorization
$$x^n - 1 = \prod_{d | n} \Phi_d(x)$$
where $\Phi_d$ is the $d^{th}$ cyclotomic polynomial. It is not hard to see that the roots of $\Phi_d$ over $F$ must in fact be precisely the set of elements of order exactly $d$ in $G$; in particular, there exists an element of order exactly $n$.
I've been told that the result for $\mathbb{Z}/p^n\mathbb{Z}$ is clearer if one first proves the corresponding result for the $p$-adic integers, but I'm not sure if this actually saves you any work.
