Is the ideal $(2,X+1)\subset\Bbb{Z}[X]$ prime, maximal or neither?

Question

I'm currently considering the ideal $(2,X+1)\subset\Bbb{Z}[X]$. I'm trying to figure out if this is a prime ideal, a maximal ideal, or neither. So far I have attempted to look at the quotient ring $\Bbb{Z}[X]/(2,X+1)$. I know that if this quotient ring is a field, then our ideal is maximal, since $\Bbb{Z}[X]$ is commutative. However I'm stuck on how I should be viewing this quotient ring.

Answer 1

Hint: $$\mathbf Z[X]/(2,X+1)\simeq (\mathbf Z/2\mathbf Z)[X ]\big/(X+1).$$

Answer 2

A nice thing about having an ideal given by a few generators, is that you can take successive quotients. That is, you can first take the quotient by one generator, and then the quotient by (the image of) the next generator. So $$\Bbb{Z}[X]/(2,X+1)\cong\Big(\Bbb{Z}[X]/(X+1)\Big)/(2),$$ and also $$\Bbb{Z}[X]/(2,X+1)\cong\Big(\Bbb{Z}[X]/(2)\Big)/(X+1).$$ Does either of these give you a better grip on your quotient ring?

Answer 3

The third isomorphism theorem says you can divide out by one generator at a time. So start with $\Bbb Z[x]/(2)\cong \Bbb Z_2[x]$. Now look at the ideal $(x-1)$ in this ring, and divide by that to see what you get. Or do it in the other order, if you think that's better.