I'm trying to work out a proof that the ring $R =\{a + b\omega : a, b \in \mathbb{Z} \}$, such that $\omega = (-1 + i\sqrt{3})/2$, is isomorphic to $\mathbb{Z}[x]/(x^2 +x +1)$. Constructing a homomorphism of the form $f:\mathbb{Z}[x] \rightarrow \mathbb{C}$ is supposed to aid me, but I'm really struggling to see why.
I'm guessing it's something to do with the isomorphism theorem and that $ker(f) = x^2 + x + 1$ is a requirement of the construction of the isomorphism $f$ but I'm just at a loss as to if this is correct and if so how to proceed.
Thanks in advance for any help.