I would like to construct a field with 729 elements. I know that $729 = 3^6$, and that I have to find an irreducible monic polynomial of degree 6 over $GF(3)$. I chose the polynomial $x^6 + 2x^2 + 1$, which I have verified (computationally) that it is irreducible over this field. However, I do not know how to proceed with the construction. Any hints?
Suppose I hadn't found this polynomial computationally. Is there an algebraic method of finding such polynomials when the degree is quite large? Or, at least testing for irreducibility in such cases?