Hi, I've problem with the following:

Let k=GF(4) and f:k^2\rightarrow k^2 is a polynomial map defined by f=(f_0,f_1), where

f_0(x_0,x_1)=x{_0}{^2}+x_1+1, f_1(x_0,x_1)=x_{1}^{2}+x_0x_1+1 and g(y)=y^2+y+a^2 is an irreducible polynomial of degree two with coefficients in k

K=k[y]/(g(y)).

\Phi:k^2\rightarrow K; \Phi(x_0,x_1)=x_0+x_1y=X

\Phi^{-1}:K\rightarrow k^2; \Phi^{-1}(X)=\left[\begin{array}{c}X \\ X^4\end{array}\right]\left[\begin{array}{cc}1+y & 1 \\ y & 1\end{array}\right]=\left[\begin{array}{c}x_0 \\ x_1\end{array}\right]

How can I show that F=\Phi\circ f\circ \Phi^{-1}
is given by
F(X)=yX^8+yX^5+X^4+X^2+X+y+1
and that
F(X)=y(X+y)(X^3+X^2+1)(X^4+(y+1)X^3+aX^2+(ay+1)X+a  ^2)
is a factorization of F(X)?

Thanks for any help.
Regards