Solving an equation in a field.

Question

I need to know the way to solve equations like this: $$(x^2+1)f(x) = 1 \pmod{x^3+1}$$ over a field $F_{3}[x]$.

Thanks in advance for any help.

Answer 1

Hint: Any residue class modulo $x^3+1$ has a unique representative of the form $a(x)=a_0+a_1x+a_2x^2$ with $a_0,a_1,a_2\in F_3$. Your task is to find the values of those constants in such a way that $a(x)(x^2+1)$ leaves remainder 1 modulo $x^3+1.$ You can brute force this either by testing all 27 combinations, or you can replace it with a linear system of three equations in the three unknowns, or you can use Calvin Lin's hint.