prove that $(1+pt)^{p^{r-1}} \equiv 1 \pmod {p^r}$

Question

I need to prove that $(1+pt)^{p^{r-1}} \equiv 1 \pmod {p^r}$

the original question is this:

Prove that , any primitive root $r$ of $p^n$ is also a primitive root of $p$

and I'm following the second answer there. I'm trying to use the binomial theorem and having hard time..

any help will be appriciated

Answer 1

Here's a good lemma to try to prove. If $s\geq 1$ then:

$$\left(1+kp^s\right)^p\equiv 1\pmod {p^{s+1}}$$

This will let you prove the result above by induction.