Let $\omega$ be a complex number such that $\omega^p=1$. Show that $$\mathbb{Z}[\omega]=\{a_0+a_1\omega+a_2\omega^2+...+a_{p-1}\omega^{p-1}|a_i\in\mathbb{Z}\}$$ is an integral domain.
The problem also came with a hint: $\omega^p=1$ implies $\omega^{p+1}=\omega, \omega^{p+2}=\omega^2,$ etc.
By the definition of integral domain, I know I need to show that if $a,b\in\mathbb{Z}[\omega]$ and $ab=0$ then $a=0$ or $b=0$.
Let $a=a_0+a_1\omega+a_2\omega^2+...+a_{p-1}\omega^{p-1}\neq0, b=b_0+b_1\omega+b_2\omega^2+...+b_{p-1}\omega^{p-1}\neq0\in\mathbb{Z}[x]$. Then $ab=a_0b_0+(a_0b_1+b_0a_1)\omega+(a_1b_1+a_0a_2+b_0a_2)\omega^2+...+a_{p-1}b_{p-1}\omega^{p-1}$. Then I need to show that this is nonzero. This is where I am stuck. I also don't think that I am on the right track because I don't know where to use the hint.