TITLE: How can I show some rings are local.
QUESTION [5 upvotes]: I want to prove $k[x]/(x^2)$ is local. I know it by rather a direct way: $(a+bx)(a-bx)/a^2=1$. But for general case such as $k[x]/(x^n)$, how can I prove it?
Also for 2 variables, for example $k[x,y]/(x^2,y^2)$ (or more higher orders?), how can I prove they are local rings?

REPLY [2 votes]: There are several other answers here, but you'll get another one.
Basic commutative algebra fact: the set of nilpotents $N$ is an ideal, and is equal to $\cap_{\mathfrak{p} \, \mathrm{prime}} \mathfrak{p}$, the intersection of all prime ideals in your ring.
Well, $x$ is nilpotent in your ring, so every prime ideal contains it(and so every maximal ideal contains it), but $(x)$ is maximal! Thus our ring is local.