Question on Galois Fields

Hi C'Punks. Here's a rather mathematical question. Can anyone explain or give an example of how to use arithmetic in GF(q^n)? Often in cryptography we work in GF(p). I knew the existence of other fields, like elliptic curves and so, but I found a short comment in Applied Cryptography page 210 that I couldn't understand. Thanks in advance for your help. Jaime acceso2@diatel.upm.es