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?