Polynomial Ring theorem proof If $p$ is a prime, prove that $(x − a)\mid (x^{p−1} − 1)$ in $Z_p[x]$ for all nonzero $a$ in $Z_p$. Hence prove that
$x^{p-1}−1=(x−1)(x−2)···(x−p+1)$ in $Z_p[x]$.
I originally thought that I could prove these using the Rational Roots Theorem but I couldn't figure out how to do it. Any help would be great, thank you in advance!

The little fermat theorem implies that $a^{p-1}=1$, this implies that $a$ is a root of $x^{p-1}-1$, if you divide $x^{p-1}-1$ by $x-a$ you obtain $(x-a)p_a(x)+b$ this implies $b=0$.
The polynomial $(x-1)...(x-p+1)$ divides $x^{p-1}$ and have degree $p-1$ the euclidean division implies that $(x-1)....(x-p+1)=c(x^{p-1}-1)$ since the coefficient of degree $p-1$ of $x^{p-1}-1$ and $(x-1)...(x-p+1)$ is $1$ you deduce that these polynomials are equal.
