Find the associated primes of $\mathbb{C}[x]/{\langle x^{3}+1 \rangle}$

Question

This intuitively to me seems to be polynomials of degree 2 or less. My idea was to show that $\langle x^{3}+1\rangle$ is primary and as $\mathbb{C}[x]$ is Noetherian we have the assassin as $\text{Ass}(\mathbb{C}/\langle x^{3}+1\rangle) = \lbrace\sqrt{\langle x^{3}+1\rangle}\rbrace =\lbrace\langle x^{3}+1\rangle\rbrace$. But this is wrong (don't know why; as an extra question why?). I then thought about trying to find the primary decomposition for the ideal but got stuck.

Answer 1

Here $$x^3+1=(x+1)(x+\omega)(x+\omega^2)$$ where $\omega=\exp(2\pi i/3)$. As ideals, $$\left<x^3+1\right>=\left<x+1\right>\cap\left<x+\omega\right>\cap\left<x+\omega^2\right>$$ and so the associated prime ideals are $\left<x+1\right>$, $\left<x+\omega\right>$ and $\left<x+\omega^2\right>$.