Mathematics
The Structure of Finite Fields and Their Applications
Quick fact
Every finite field has exactly p^n elements, where p is a prime and n is a positive integer. The most famous example is GF(2^8) used in AES encryption and Reed–Solomon error correction codes, which operate byte-wise on this 256-element field.
Why this is interesting
You've probably dealt with 12-hour clocks and the wrap-around of modular arithmetic. But what if you could build an entire number system with a finite set of numbers, where every operation still works perfectly?