TITLE: Finite Fields. Why does it involve prime numbers only?
QUESTION [6 upvotes]: I'm just getting my head around the finite fields, so called Galois Fields. Why are they based on prime numbers only? any concept I'm missing?

REPLY [2 votes]: Let's try it with a composite number, $12$.  We have $3\cdot 4 = 0$, so
$$
0 = 3^{-1}\cdot 0 = 3^{-1}\cdot(3\cdot4) = (3^{-1}\cdot3)\cdot 4 = 1\cdot4 = 4.
$$
That's why it does not work with composite numbers.
Why it does work with prime numbers is a more substantial thing: How do you prove everything except $0$ has a multiplicative inverse in that case?  That involves the quotients in Euclid's algorithm.