Applications of GF(p) polynomials
In my knowlegde, XTR is a cryptosystem build over $GF(p^6)$. We however use the subfield $GF(p^2)$ to have a compact representation.
I don't know if there are other cryptosystems defined directly on Galois field extention, but using some tools are common. For that, take a look on the operative environment of Weil or Tate pairing.