Show that the order of the class of $p+1$ in $\left( \mathbb{Z}/p^{\alpha}\mathbb{Z}\right)^{*}$ is $p^{\alpha-1}$

Question

I tried to do that:

$$(1+x)^p=1+px+\binom{p}{2}x^2+\dots+\binom{p}{p-1}x^{p-1}+x^p$$

So if $x=0 \quad mod(p^k)$ then $(1+x)^p=1 \quad mod(p^{k+1})$

Now I'm trying to deduce that $(1+p)^{p^{\alpha-1}}= 1 \quad mod(p^\alpha)$ but I don't know how to do it. Maybe recurrence?

Thank you