is it possible to have field $\displaystyle GF(((2^2)^2)^2)$ is constructed from the following irreducible field polynomials:
$\displaystyle r(z) = z^2 + \{0100\}z + 1$ for $\displaystyle GF(2^8)/GF(2^4)$
$\displaystyle s(y) = y^2 + \{10\}y + 1$ for $\displaystyle GF(2^4)/GF(2^2)$
$\displaystyle s(x) = x^2 + x + 1$ for $\displaystyle GF(2^2)/GF(2)$