[tjade273]

A set of numbers with addition and multiplication is a _ring_ and is a _field_ only if every nonzero number has a multiplicative inverse.

Integers mod N for composite N (like in RSA) form a ring, not a field - if N = p*q then there is is no well-defined division by `p` or `q` (for example, there is no element corresponding to `2/p`).

When p is prime, every nonzero integer `x` has a multiplicative inverse `1/x` mod p. That's why the integers mod `p` form a field (denoted F_p).

In fact there is another kind of finite field that has `p^n` elements where `p` is any prime and `n` is any positive integer. These fields are not composed of the integers mod p^n, but are made of polynomials of degree `n` with coefficients in F_p.