Question: Find $\gcd$ of $x^4+3x^3 +2x+4$ and $x^2-1$ in $\mathbb{Z}_5[x]$
Applying the Euclidean Algorithm as my book suggests, I got the following:
$x^4+3x^3+2x+4=(x^2-1)(x^2+3x+1)+(5x+5)$
$x^2-1=(5x+5)(\frac{1}{5}(x-1)) +0$
However we see that $5x+5$ is not monic. Book says "In that case, multiply it by the inverse of its leading coefficient to obtain the gcd". But in $\mathbb{Z}_5$, $5=0$, so where is the inverse? Thanks!
Just to make it official, the kicker is that $5\equiv0\pmod5.$