Irreducible quadratics in polynomial ring of two variables over algebraically closed field I'm currently stuck at problem 1.1 c) in Hartshorne's algebraic geometry book. I just can't let it go. Setting is as title says (field $k$, variables $x$ and $y$).
Problem 1.1. a) and b) concerns themselves with the special cases $y-x^2$ and $xy - 1$, and classifying the resulting quotient rings (being isomorphic to a polynomial ring in one variable over $k$ in both cases, but allowing negative exponents in the second).
c) asks of me to prove that these are the only two possible outcomes, up to isomorphism. And I just can't. Any help would be appreciated.
A related question I came up with was, if we're in case b), since any element in $k$ can be inverted, and x can be inverted, then surely, any element in the ring can be converted, and we have a field. Is this so?
 A: An irreducible quadratic corresponds to a nonsingular conic in $\mathbb{P}^2_k$. 
Choose three points on the conic, and select projective coordinates so that these points are $(1:0:0)$, $(0:1:0)$, and $(0:0:1)$. In these coordinates, the conic has equation $cxy + ayz+bxz=0$ for some nonzero $a,b,c$ (if any of $a$, $b$, or $c$ are zero, then the conic is singular). Rescaling by multiplying $x$ by $a$, $y$ by $b$, and $z$ by $c$ gives that the conic can be written with equation $xy+yz+xz = 0$.
This conic is isomorphic to the projective line $\mathbb{P}^1_k$.
The regular functions on the conic $xy+yz+xz=0$ minus some hyperplane are therefore equivalent to regular functions on $\mathbb{P}^1_k$ minus one or two points. The regular functions on the projective line minus a single point yield $k[x]$, and the regular functions on the projective line minus two points yield $k[x,x^{-1}]$. You get the latter if and only if the conic intersects the line at infinity in exactly one point. 
Added. I missed your "related" question. No, it is not true that $k[x,x^{-1}]$ is a field: it's the ring of Laurent polynomials with coefficients in $k$; i.e., expressions of the form $$a_{-m}x^{-m} + \cdots + a_{-1}x^{-1} + a_0 + a_1x + \cdots + a_nx^n$$ for some nonnegative integers $m$ and $n$. But this is not a field. For instance, $x-1$ cannot be inverted in $k[x,x^{-1}]$.
