Are there any other constructions of a finite field with characteristic $p$ except $\Bbb Z_p$? I mean, $\Bbb Z_p$ is an instance of $\Bbb F_p$, I wonder if there are other ways to construct a field with characteristic $p$?
Thanks a lot!

There are also extensions of $\mathbb F_p$.  For example $\mathbb F_2[t]/(t^2 + t + 1)$ is a field with $4$ elements.  It has characteristic $2$.
You might try perusing the wikipedia page which has more examples.
