A ring and a quotient field with finite characteristic Lacking imagination, for understanding purposes I would like to see an example of an integral domain (with unity) that is not a field but has a quotient field of finite characteristic. If convenient an examples with finite and infinite quotient fields are appreciated.

The ring of polynomials $\mathbb{F}_p[T]$ has a fraction field of $\mathbb{F}_p(T)$, which is of characteristic $p$. In fact, any $\mathbb{F}_p$-algebra that is an integral domain will have a fraction field of characteristic $p$.
There will not be any examples of non-field integral domains whose fraction field is a finite field, because this would imply that the original integral domain was finite,  and any finite integral domain is already a field.
