## Irreducible Polynomial Help

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

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