I am sorry. But I wonder whether it is fine to ask this question here.
I need some help to solve some equations in a polynomial ring:
Give the ring $\mathbb{F}_p [x ]/(x^2+x+1)$, where $p$ is a prime (for example $p=5$). I need to solve some equations like:
$(1-x)f_i(x)+x f_j(x)-f_k(x)=0$
I have no idea how to solve this in Mathematica. (Of course, we can solve them via pencil and paper. But it is a headache if it involves 20 or more equations.) Any help is appreciated.
My own approach was to set $f_i(x) = a+b x$ with $a,b \in \mathbb{F}_5$. Then try to solve the (linear) equations $mod\ 5$. But it failed because we may need $x^2+x+1=0$.