I had a curious question bouncing around my head the other day. I asked myself if there were numbers with similarly nice properties to something like $e$ in a finite field, along the lines of $\mathbb{F}_p$. But because these finite fields tend to lack the structure of continuity of something like $\mathbb{R}$ I started to think about an extension by using the interval $[0,p] \subset \mathbb{R}$ with the added structure that $p \equiv 0$ (I was thinking about cutting a segment out of the real number line and glueing it together at $0$ and $p$). Would that be a well-defined structure somewhat analogous to a like "pseudo"-finite field? And if that structure is a thing, how would one go about defining the analytic tools needed (for example a limit) to define interesting properties like $\frac{de^x}{dx}=e^x$ or $cos(\pi) = 0$.
I am only in my second semester in the subject so I am not too familiar with advanced mathematics, and I would be really curious how these operations would behave in these circumstances or if this even can logically be a thing.