Find a inverse of polynomial in quotient ring in $\mathbb{Z}_p$

Question

Let $f=x^3+x+1$, I want to find a inverse of $g=x^4 +3x^3+x^2+1$ in $A=\mathbb{Z}_5[x]/(f)$.

I do the division between $f$ and $g$ is $q=x+3$ and $r=2x+3$, then I continued with the division between $x^3+x+1$ and $r$ and then I have rest $=1$.

Now I don't know how to move in order to find the inverse.

Thanks for the attention.

Answer 1

Represent the gcd of the two polynomials as a linear combination of these polynomials using the extended euclidean algorithm (Bezout's thm):

$1 = a\cdot f + b\cdot g$ for some polynomials $a,b$.

Then $b\cdot g\equiv 1 \mod f$ and so $b\cdot g =1$ in $\Bbb Z_5[x]/\langle f\rangle$.