A field isomorphism related to polynomial rings and their field of fractions There are 2 ways to approach function fields: the algebraic approach, i.e. looking at finite extensions of $K(s)$, where $s$ is transcendental. The other is geometric, i.e. considering functions over a curve: $K[s,t]/(q)$ where $q \in K[s,t]$ represents a plane curve. 
My question is - are these 2 approaches basically the same? 
Specifically: 


*

*Is $K(s)[t]/(q)$ naturally isomorphic to $K[s,t]/(q)$?

*Can every finite extension of $K(s)$ be represented as $K[s,t]/(q)$ for some $q$?

 A: The two approaches are basically the same, but you're missing some details.
I believe the relevant theorem is:
There is a one-to-one correspondence between isomorphism classes
of finite extensions of K(s) and isomorphism classes of complete,
non-singular curves over K

I couldn't find a reference, so I may have some small technical detail wrong on what sort of isomorphism classes; e.g. maybe the correspondence involves isomorphism classes of extensions $E/K$. Or maybe it's between isomorphism classes of extensions $E/K(s)$ with isomorphism classes of bundles $X \to \mathbb{P}^1_K$? 
The correspondence sends a curve $X$ not to its coordinate ring $\mathcal{O}(X)$, but to its function field $K(X)$, which is the fraction field of $\mathcal{O}(X)$. The reverse correspondence, IIRC, can be constructed by considering the discrete valuations of the field.
In the case of an affine plane curve $X$ defined by $f(s,t) = 0$, where $f$ is irreducible and not contained in $K(s)$, we have:


*

*The ring of coordinate functions is $\mathcal{O}(X) \cong K[s,t] / f(s,t)$

*The field of rational functions is $K(X) \cong K(s)[t] / f(s,t)$


One can say similar things with curves in higher dimensions; e.g. the curve in three dimensions defined by a pair of functions of three variables. If $X$ is non-singular, then $\mathcal{O}(X)$ is a Dedekind domain. The theory of curves, in fact, is extremely closely related to the theory of algebraic number fields -- localizations of the ring of integers of an algebraic number field are also Dedekind domains.
Note that the aforementioned correspondence tells us there's a unique way to modify our affine curve $X$, extending it to a complete curve and desingularizing it.
Every finite algebraic extension of a field can be obtained by iterating the construction
$$ K \mapsto K[t] / f(t) $$
whether or not you can do it in just one step is the subject of the primitive element theorem. Doing it in $n$ steps merely corresponds to a curve in $n+1$ dimensions.
