How are field representations chosen?
For polynomial basis, one looks for trinomial reduction polynomials to improve the speed of modular reduction. A trinomial is a polynomial with just three terms, for example x^29 + x^2 + 1. For normal basis, one looks for an ‘optimal’ normal basis such that field multiplication is efficient. For hardware, a normal basis over GF(2) may be attractive. For software, a polynomial basis, or a sub-field representation seems to be better (more efficient). In principle, one may convert between different field representations, so looking for a field in which several different efficient representations are possible might be a good idea. One possible candidate is GF(2^261) which can be represented with an optimal normal basis, and also as a polynomials with coefficients in GF(2^9) using the trinomial x^29 + x^2 + 1 as the reduction polynomial.
