Q:

What are the k-rational points of k[[t]]?

Let $k$ be a field. What are the $k$-rational points of the affine $k$-scheme $\mathrm{Spec}(k[[t]])$, where $k[[t]]$ is the power series ring over $k$ (equivalently, what are the $k$-algebra morphisms $k[[t]] \rightarrow k$?)
I'm only sure about one point, namely the map $t \mapsto 0$. Do I have to assume some sort of completeness of $k$ to get more points? 
Is there a nice presentation of $k[[t]]$, i.e. a quotient of some polynomial ring that is isomorphic to $k[[t]]$?

A:

$k[[t]]$ is a local ring with maximal ideal $(t)$ and the kernel of every $k$-homomorphism $k[[t]] \to k$ is a maximal ideal, thus the maximal ideal. Thus it factors as $k[[t]] \to k[[t]]/(t) = k \to k$ and $t \mapsto 0$ is the unique $k$-rational point.

