A Galois field of the type $GF(p)$, where $p$ is prime, is normally expressed as the ring of integers modulo $p$.
If my understanding is correct, it is also possible to represent its elements as a ring $\mathbb Z_2[X]$ of polynomials defined over $GF(p)$, i.e. for $GF(2)$, we'd have polynomials of the form similar to $x^8+x^5+x^3+1$ or $x^9+x^6+x^2$, etc, and for $GF(5)$, we'd have polynomials of the form similar to $4*x^7+2*x^3+3*x+1$ or $2*x^5+3*x^3+x^2$, etc. All arithmetic operations are like with normal polynomials, but with a difference: all arithmetic operations on coefficients which occur in the calculations are done modulo $p$.
The question is: do polynomials defined over $GF(p)$ have any real application in cryptography where they are a better fit than their integers-modulo-$p$ counterparts?
Please correct me if I mixed something up in the theoretical part!