Let $\mathbb M$ be one of the following rings: $\mathbb{R}, \mathbb{Q}, \mathbb{F_9}, \mathbb{C} $. Let $I$ be the ideal generated by $x^4+2x-2$. Is the ring $\mathbb M[x]/I$ a field for some of the above choices for $R$?
I have been trying to work my way around some problems involving ideals, but I get really confused when they get mixed with rings and fields. I'm stuck and don't know how to proceed.
In general, given a ring $R$ and an ideal $J$ of $R,$ we have that $R/J$ is a field if and only if $J$ is a maximal ideal of $R.$ Now, since each $\Bbb M$ is a field, here, then $\Bbb M[x]$ is a principal ideal domain, and so $I$ is maximal if and only if $I=\langle p(x)\rangle$ for some irreducible polynomial $p(x)\in\Bbb M[x].$ So, what you must determine is whether $x^4+2x-2$ is irreducible over any of the given fields $\Bbb M.$
Added: As a hint for how to address the case that $\Bbb M=\Bbb F_9,$ note that $\Bbb F_9$ has characteristic $3,$ and so $x^4+2x-2=x^4+2x+1,$ as far as $\Bbb F_9$ is concerned.