Use Bézout’s Identity to generate the inverse: if in the Bézout’s identity:
\[(a.x) + (b.y) = gcd(a, b)\]we place
\[a.x = f(x).f'(x) \\\] \[b.y = a(x).(x^N -1)\]such that the GCD is 1.
hence we get,
\[f(x).f'(x) + a(x).(x^N -1) = 1\]here we have
\[a(x).(x^N -1) = I\]where $I$ is the ideal of the Quotient Ring that we are working with. As Ideal are congruent to 0 in a quotient Ring we may simply write:
\[f(x).f'(x) = 1\]thus meaning $f’(x)$ is the inverse of $f(x)$.
[!Note] To find inverse of the polynomial we have to use the Bézout’s identity. Where the Euclidean algorithm will be used on $f(x)$ and $x^N - 1$ with $GCD = 1$ thus giving us the values for $f’(x)$ and $a(x)$