I am working with equations in a finite field extension and I am looking for a mathematical software to solve the following problem.
Let $k$ be a field $GF(4)$. I have a function $f:k^2 \to k^2$ defined via $f = (f_0,f_1)$, where each $f_i$ is a bivariate polynomial in $k[x,y]$. Let $g(z) = z^2+z+a \in k[z]$ be an irreducible polynomial; Here $a$ is the generator of the group of nonzero elements of $GF(4)$. Define $K= k[z]/(g(z))$ a field extension of $k$. I also have a map $\phi:k^n \to K$ via $\phi(x,y) = x+yz = X$, and a function $F(X) = \phi \circ f \circ \phi^{-1} (X)$. The inverse of $\phi$ is also given and it yields $\phi^{-1}(X) = (x,y)$.
I can write $F(X)$ in terms of $x,y$ and $z$ using the formula $F(X) = \phi \circ f \circ \phi^{-1} (X)$. However, I want to write it in terms of $X$, keeping in mind that I work in $K = k[z]/(g(z))$. I was wondering if there is any mathematical software available to do that? Thank you!