In galois field of prime 2, in composite field $GF((({2}^2)^2)^2)$,
There are irreducible polynomials and reducible polynomials.
$GF(2^2):Q_1(x) = x^2+x+1,$
$GF((2^2)^2):Q_2(x) = x^2+x+\phi,$ $\alpha$ is the root of $Q_2(x)$, $\phi \in GF(2^2)$
$GF((2^2)^2)^2):Q_3(x) = x^2+x+\lambda,$ $\lambda \in GF((2^2)^2)$
Combinations of $\phi$ and $\lambda$ constructs the field.
I tried to figure out the combination $\phi=\{10\}$ and $\lambda=\{1010\}$ is reducible $(\lambda =\alpha^9$ at $GF((2^2)^2))$, but cannot find the pair that makes polynomial $Q_3(x)$ reducible.
Is there any pair that makes this polynomial reducible?
$\phi$ can be expressed as $\{1\}X^1 + \{0\}X^0$
$\lambda$ can be expressed as $(\{1\}X^1+\{0\}X^0)Y^1 + (\{1\}X^1+\{0\}X^0)Y^0$
Element of $GF((2^2)^2)^2)$ can be expressed as
$((a_7X^1+a_6X^0)Y^1+(a_5X^1+a_4X^0)Y^0)Z^1+((a_3X^1+a_2X^0)Y^1+(a_1X^1+a_0X^0)Y^0)Z^0$,$\{a_7 a_6 a_5 a_4 a_3 a_2 a_1 a_0\}$.
Representation is different to $GF(2^8)$