Prove or disprove that $\mathbb{Z}[x] / (x^3 - 2)$ is a field. Prove or disprove that $\mathbb{Z}[x] / (x^3 - 2)$ is a field.
I am pretty sure it is not a field because if you consider $F = \mathbb{Z}[x] / (x^3 - 2)\mathbb{Z}[x]$ where $F$ is a field, then char $F = 0$.
Not sure how to write a proof for this though. 

Hints:
$$\begin{align*}\bullet&\;\;\;x^3-2\in\Bbb F_7[x]\;\;\text{is irreducible}\\{}\\
\bullet&\;\;\text{As ideals in}\;\Bbb Z[x]\;\;,\;\;\langle\;x^3-2\;\rangle \lneqq\langle\;x^3-2\,,\,7\;\rangle\end{align*}$$
You may want to try to prove the general theorem: an ideal in $\;\Bbb Z[x]\;$ is maximal iff it is of the form $\;\langle f(x),p\rangle\;$ , with $\;p\;$ a prime and $\;f(x)\in\Bbb Z[x]\;$ such that $\;f(x)\pmod p\;$ irreducible in $\;\Bbb Z_p[x]\;$ .
Personally for me, the above is one of the most stunning and beautiful facts in basic abstract algebra.
