number theory - What does a solution of a Diophantine equation in $mathbb{F}_{p}$ tell about general integer solution?

Q: What does a solution of a Diophantine equation in $\mathbb{F}_{p^k}$ for all $k\in\mathbb{N}$ and prime $p$ tell about general integer solution?

We know that if there are not any solutions of a Diophantine equation in $\mathbb{F}_p$ then there are not any solutions in integers. Is this the only reason to study solutions in $\mathbb{F}_p$ or there are other reasons?

I read these two Wikipedia articles 1 and 2 but couldn't find answer to this question.