Is the ring $K=\mathbb R[t] /( t^3+1)$ integral domain and/or a field? What's $[K: \mathbb R]$?

Question

Is the ring $K=\mathbb R[t] / (t^3+1)$ integral domain and/or a field? What's $[K:R]$?

Since $t^3+1$ is reducible over $\mathbb R$, $K$ isn't a field. How do I check whether it's an integral domain?

Also, how do I calculate this (bizarre?) degree $[K: \mathbb R]$? Maybe it's a typo and should be $[K:\mathbb R[x]]$? How do I calculate it then?

Thanks in advance for any assistance!

Answer 1

The ring is not an integral domain. Since $t+1$ and $t^2-t+1$ are non zero elements, however their product is $t^3+1=0$.