I am currently trying to find a primitive element of the multiplicative group of field $GF(p)$. Since the numbers are relatively small, I know the factorization of
$\phi(p)=p-1 = {p_1}^{k_1} {p_2}^{k_2} ... {p_n}^{k_n}$
Wikipedia says that $m\in GF(p)$ is a generator if
$m^{\phi(n)/p_k} \not\equiv 1 \mod p \quad \forall k=1..n$
However, there is no clear explanation why it is the case. Could you please help me to understand this?