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!

For each power $p^r$ of $p$, there is a unique finite field of characteristic $p$ and size $p^r$, it is denoted by $\mathbb F_{p^r}$.
For the construction, you take an irreducible polynomial $f\in\mathbb F_p[x]$ of degree $r$. Then $\mathbb F_{p^r} \cong \mathbb F_p[x]/(f)$.
The smallest non-trivial example is $\mathbb F_4 = \mathbb F_{2^2}$. A concrete construction for it is given in the answer of Jim.
