In the field $\mathbb{Z}_7[x]/\langle x^4+x+1\rangle$, find the inverse of $f(x)=x^3+x+3$. In the field $\mathbb{Z}_7[x]/\langle x^4+x+1\rangle$, find the inverse of $f(x)=x^3+x+3$.
I know how to find the inverses of elements within sets, rings, and fields. I know what to do if the field was just $\mathbb{Z}_7$, but the fact that the field is $\mathbb{Z}_7[x]/\langle x^4+x+1\rangle$ confuses me. I don't know where to start.

I would guess that you would first check that $\gcd(x^{4}+x+1,x^{3}+x+3)=1$ in $\mathbb{Z}_{7}[x]$ (else $x^{3}+x+3$ is a zero divisor in the field, thus cannot have an inverse).
Then use the characterization of the gcd as a linear combination to write $$p(x)(x^{4}+x+1)+q(x)(x^{3}+x+3)=1.$$ in $\mathbb{Z}_{7}[x]$. When we project down to the specified field, we kill the $x^{4}+x+1$ term (by the definition of the field), so the desired inverse is the projection of $q(x)$.
