$\frac{\mathbb Z[x]}{(x^2)}$ is a local ring?

Question

I would like to prove if $\frac{\mathbb Z[x]}{(x^2)}$ is local ring. I think it do is a local ring, since $x^2 $ might be a maximal idea, but I do not how to express it, since I am trying to explore localization by myself. So I will really appreciate any help. Thank you so much.

Answer 1

I like thinking of maximal ideals as maps to fields (this is really just a restatement of @KCd's comment). Any map out of $\mathbb{Z}[x]/(x^2)$ is determined where $1$ and $x$ are sent. Can you think of any maps from this ring to, say, a finite field? Is the size of the finite field relevant?