How to decide if the multiplicative group of non-zero elements of a finite field is cyclic or not?
Based on our experience with $\left(\mathbb{Z}_7 \setminus \{0\}, \cdot \right)$, which is generated by $3$ or $5$, and with $\left(\mathbb{Z}_{11} \setminus \{0\}, \cdot\right)$, which is generated by $2$, $6$, $7$, or $8$, can we conclude that $\left(\mathbb{Z}_p \setminus \{0\}, \cdot\right)$, where $p$ is any prime, is cyclic? And if so, then which elements can generate this group?