Originally Posted by

**jamix** I'm given the field $\displaystyle Z_2[t]/(t^3 + t + 1)$ for which I need to calculate the inverse of each element.

I assume the way to do this is similar to the Extended Euclidean Algorithm and this is what I have for computing the inverse of $\displaystyle t^2 + 1$. I'm a little bit skeptical with my reasoning however and this is partially due to my lack of familiarity of working with polynomials over $\displaystyle Z_p[x]$.

1) First use the Division algorithm to obtain $\displaystyle t^3 + t + 1 = t*(t^2 + 1) + 1$

From the above, this gives us $\displaystyle -1*(t^3 + t + 1) + 1 = -t*(t^2 + 1)$.

On the other hand, as we are working over $\displaystyle Z_2[t]$, it follows that we must have $\displaystyle (t^3 + t + 1) + 1 = t*(t^2 + 1)$ since $\displaystyle -1 \equiv 1$. Hence the inverse of $\displaystyle t^2 + 1$ is just $\displaystyle t$.

Is this correct?