Prove that the rings $F[x,y]/(y^2-x)$ and $F[x,y]/(y^2-x^2)$ are not isomorphic for any field $F$.
I suppose I should set up an arbitrary supposed isomorphism between them and show that a contradiction arises, but I'm not really sure on the details of the implementation. Thanks for any help.
Hint: In the latter ring you have zero divisors, as $y^2 - x^2 = (y - x)(y + x) = 0.$ Now, express the former ring in a more familiar presentation by eliminating a variable. Remember that in this quotient, $y^2 - x = 0.$