let say i have $\displaystyle P_1(x)= x^2+ x + \alpha$ as irreducible polynomial over $\displaystyle GF(2^2)$ to construct $\displaystyle GF((2^2)^2)$. and $\displaystyle P_2(x) = x^2+ x + \beta $ as irreducible polynomial over $\displaystyle GF((2^2)^2) $ to construct $\displaystyle GF(((2^2)^2)^2)$.

if i have $\displaystyle \alpha = \{10\}_2$ or $\displaystyle \{11\}_2$ , what are the possible value for $\displaystyle \beta$ in order to be irreducible??