I am supposed to find the inverse of $1 + x^2$ in the division ring $\mathbb{ Z }_{ 11 }[ x ] / \langle x^3 + 1 \rangle$.
I know that that $x^3 = -1 = 10$. I have tried solving for $a$, $b$, and $c$ in $( ax^2 + bx + c )( x^2 + 1 )=1$, but I have found no obvious answer (I've found $a = b = -1/2$, $c = 1/2$, but that obviously doesn't help).
Is there some simple way of doing this that I am missing?