Suppose I have the following system of equations:
\begin{cases} A+B+C+D+P+Q=0 \\ Ax^5+Bx^4+Cx^3+Dx^2+Px+Q=0 \end{cases}
Also suppose that we are working in a $ GF(2^4) $ field.
Use the following irreducable polynomial: $g(x)=x^4+x^3+1$
How can I use the gauss elimination algorithm to find P and Q in function of the other letters?
Subtract the first equation from the second and you would get:
$$P(x-1) = A(1-x^5) + B(1-x^4) + C(1-x^3) + D(1-x^2)$$
Now find the inverse of $(x-1)$ in $GF(2^4) \cong \mathbb{Z}_2/\langle g(x) \rangle$ and multiply both sides to find out the value of $P$. As you should probably now you can use the Euclidean Algorithm to find out the inverse.
Afterwards directly from the first equation you get $Q = -A -B -C -D -P$, where you will substitute the value of $P$ from above.