Existence of primitive roots

Finite abelian group. $\mathbb{F}_p^*$ is a finite abelian group of order $p-1$.

Structure theorem. Any finite abelian group decomposes as $\prod_i\mathbb{Z}/n_i\mathbb{Z}$ with $n_i\mid n_{i+1}$. The exponent is $n_r\mid p-1$, and every element satisfies $x^{n_r}=1$, so $n_r\ge p-1$ (since the polynomial $X^{n_r}-1$ has at most $n_r$ roots in the field). Thus $n_r=p-1$, and the group is cyclic.

Conclusion. $\mathbb{F}_p^*$ is cyclic of order $p-1$, so primitive roots exist.