My background is in computer engineering, not pure math. I've been studying the application of Finite Fields. Wikipedia on Finite field arithmetic says:
GF(p), where p is a prime number, is simply the ring of integers modulo p.
Elements of GF($p^n$) may be represented as polynomials of degree strictly less than n over GF(p).
I would really appreciate a simple, accessible explanation as to why this is the case? What fundamental property of Finite Fields makes it that the elements of $GF(p)$ and $GF(p^n)$ behave so differently and yet are both considered Finite Fields?