TITLE: the the polynomial $K$-algebra $K[t_1, t_2]$ is not local
QUESTION [5 upvotes]: Let $A$ be the polynomial $K$-algebra $K[t_1, t_2]$. Then the algebra $A$ is not local, why? More generally, $K[t_1, t_2, \cdots, t_n]$ is not local for any $n\geq 1$, why?

REPLY [4 votes]: Seems like the easiest way to look at all of them is this way: $t_1$ and $1-t_1$ aren't units, so they generate proper ideals $(t_1)$ and $(1-t_1)$. If there were a unique maximal proper ideal containing both, it would have to contain $t_1+(1-t_1)=1$, but that would contradict the fact the ideal is supposed to be proper.

REPLY [2 votes]: How about the fact that you can find two distinct maximal ideals of $K[t_1, t_2]$? Like $(t_1,t_2)$ and $(t_1 - 1, t_2 - 1)$, for instance. In general, $K[t_1, \dots, t_n]$ is not local because of the ideals 
$$
(t_1, \dots, t_n) \quad \text{ and } (t_1 - 1, \dots, t_n - 1).
$$
Hope that helps,