Place at infinity in function fields I've just started reading about function fields.
If I understood it correctly, for a function field in one variable $F|K$ there is a correspondence between places (i.e., maximal ideal of a valuation ring $K \subsetneq O\subsetneq F$), discrete valuations and valuation rings.
But I don't understand what people mean by "place at infinity". 
In the case of the rational function field $K(x)|K$, for example, what is the place at infinity? Which valuation ring is it associated to?
 A: The function field $K(x)$ is the field of fractions of the affine domain $K[x]$.  Most places of $K(x)$ come from maximal ideals of $K[x]$ (i.e., the associated valuation ring is a localization of $K[x]$ at a maximal ideal).  However, there's one place that doesn't: namely, the place corresponding to the localization of $K[1/x]$ at the maximal ideal $(1/x)$.  This is called the "place at infinity".  One reason for this is that the associated valuation takes a rational function $f$ to the "order of vanishing of $f$ at $\infty$" (if the numerator of $f$ has degree $a$ and the denominator has degree $b$, that's $b-a$, which intuitively measures how fast $f(x)$ approaches $0$ as $x\to\infty$).
Another explanation for the name is geometrical.  You can think of $K(x)$ as the field of rational functions on the affine curve $\operatorname{Spec} K[x]=\mathbb{A}^1$.  All of the places except the place at $\infty$ correspond to closed points in this affine curve.  The place at $\infty$, however, corresponds only to a point in its completion $\mathbb{P}^1$, where it is the usual "point at $\infty$" in the projective line.
Similarly, if $F$ is any other function field that is the field of rational functions on some smooth affine curve $C$, then places of $F$ correspond to points in the completion of $C$.  The places that correspond to actual points of $C$ are "finite places", and the places that correspond only to points in the completion are "infinite places".
Note that this notion of "infinite place" is not intrinsic: it depends on a choice of affine domain that you view your function field as the field of fractions of.  For instance, if you consider $K(x)$ as the field of fractions of $K[1/x]$ instead of $K[x]$, then what used to be the infinite place is now a finite place.  This is in contrast to the case of number fields, where infinite places really are intrinsically different from other places.
