Let $I_1$ be the ideal generated by $x^2 +1$ and $ I_2$ be the ideal generated by $x^3 – x^2 + x -1$ in $\mathbb{Q}[x]$. If $R_1= \mathbb{Q}[x]/ I_1$ and $R_2= \mathbb{Q}[x]/ I_2$ then which of the followings are true?
1. $R_1$ and $ R_2$ are fields
2. $R_1$ is a field and $ R_2$ is not a field
3. $R_1$ is an integral domain, but $ R_2$ is not an integral domain
4. $R_1$ and $ R_2$ are not integral domains.
Here $x^2+1$ is irreducible but $x^3 – x^2 + x -1$ is not irreducible. So I think $R_1$ is field but $R_2$ is not. So 2 and 3 are true. Am I right?