Prove that $(x^3-2)$ is a maximal ideal of $\Bbb Q[x]$

Question

Prove $(x^3-2)$ is maximal ideal of $\Bbb Q[x]$ using isomorphism theorems for rings.

I tried using the second isomorphism theorem for rings, to use that $( x ^ 3-2)$ is maximal if and only if $\mathbb Q [x ] / ( x ^ 3-2)$ is field . Seeing that $\mathbb Q [x ] / ( x ^ 3-2)$ is isomorphic to a field related to $\mathbb Q$. I have $A = \mathbb Q [x ]$ , $I = ( x ^ 3-2)$ but not to $B$ take to apply $( B + I) / I $ isomorphic $B / (B ∩ I )$ field.

Answer 1

The ideal $(x^3-2)$ is maximal iff $x^3 - 2$ is irreducible over $\mathbb{Q}$, which is obvious.