galois theory - Irreducible Polynomial for $GF(2^{256})$

I'm looking for a pattern to generate Galois Field multiplication for $2^{256}$ binary value. So far I have come up with a patter as follows;

$$1 \rightarrow 1 \\ x \rightarrow x \\ x^2\rightarrow x^2\\ ...\\ x^{256} \rightarrow x + 1$$

Is it $x + 1$ for $x^{256}$?
If so, for $x^{257} \rightarrow (x^2 + x)$?

Thanks in advance!