Is the quotient ring $\Bbb{Z}_7[x]/\langle 3x^3 + 4x^2 + 6x + 4\rangle$ an integral domain?

Question

Is the quotient ring $\Bbb{Z}_7[x]/\langle 3x^3 + 4x^2 + 6x + 4\rangle$ an integral domain?

I don't know where to start. I understand what it means to be in $\Bbb{Z}_7$ but do not completely understand quotient rings and integral domains.

Answer 1

The quotient ring is an integral domain if the polynomial is irreducible. It happens $2\,$ is a root of $3X^3+4X^2+6X+4=3X^3-3X^2-X-3$ in $\mathbf F_7$, so this polynomial is divisible by $X-2$, and it factors as: $$3X^3-3X^2-X-3=(X-2)(3X^2+3X-2),$$ The latter factor has no root, so it's irreducible.

Answer 2

As $(x-2)(3x^2+3x-2)=0 \mod 3x^3+4x^2+6x+4$ in $\mathbb{Z}_7(x)$ we can conclude that $\mathbb{Z}_7(x)/(3x^3+4x^2+6x+4)$ is not an integral domain.