Let $p$ be an odd prime number. The multiplicative group modulo $p^α$, $U_{p^α}$, is cyclic for all $α\geq 1$.
I have read proofs of this result in Vinogradov "Elements of number theory" (published in 1954) and in Ireland "A classical introduction to modern number theory" (1982) but could not fully understand them (see my post: Vinogradov's proof that $U_{p^{\alpha}}$ is cyclic. How to prove $p^{r-1} (p-1) \mid \delta$ for $1 \leq r \leq \alpha$).
I am really willing to understand this result, by reading other proofs of this result (be it a similar proof, written by another author, or be it another proof, i.e. another valid way of proving it).
The result seems to be a pretty classical one. However I do not know which keywords to type in the archive search bar (Archive is an online librairy) to find books covering the subject.
I've been thinking about "introduction in number theory" or "elementary group theory" but is seems to be too wide a scope.
Thank you.