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.
A: Perhaps one answer to your question about deformations is something like the following. A deformation over a complete local ring A (such as k[[t]]) is just a family X $\to$ Spec(A). Suppose that the fibers belong to some sort of moduli space M, such as the moduli space of curves. In the functorial point of view of moduli spaces, the family X $\to$ Spec(A) corresponds to a morphism Spec(A) $\to$ M that assigns to a point of Spec(A) the moduli of the fiber over this point. So, one parameter formal deformations (by this I just mean that A = k[[t]]) correspond precisely to the morphisms Spec(k[[t]]) $\to$ M. The scheme Hom(k[[t]], M) is called the space of arcs in M. If we fix the central fiber of the deformation then we get the space of arcs in M at the point corresponding to the central fiber. The space of arcs is a subtle and important invariant of a singularity. One can think of an arc (that is, a morphism Spec(k[[t]]) $\to$ M) as follows: if we had a curve in M then the arc would be the collection of jets this curve determines, ie all the derivatives of all orders of the curve (think of the way morphisms Spec$(k[t]/(t^2)) \to M$ determine the tangent vectors at the image of the closed point). So these deformations are telling us something significant about the local structure of the moduli space.
The construction of these one parameter formal deformations works regardless of the existence of any moduli space. It tells us what the space of arcs on the moduli space of whatever it is you are deforming should be.
