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. 

Why do you think char 0 matters? If you can answer that question, that would give a lead to coming up with a proof.
It does turn out to be true that every quotient field of $\mathbb{Z}[x]$ has nonzero characteristic, so there is surely a proof along these lines.  It would require two parts: showing that $\mathbb{Z}[x] / (x^3-2)$ is actually characteristic 0 (under the assumption it is a field), and showing that this is a contradiction.
I don't know what particular things you know, so I'm not really sure how to suggest going about the problem along these lines.
The simplest is probably, IMO, inspired by the fact a char 0 field must contain $\mathbb{Q}$: find some integer $n$ for which you can easily prove that $n$ is both nonzero and noninvertible in $\mathbb{Z}[x] / (x^3 - 2)$. 
