k: algebraic closed field.
A(X): affine coordinate ring of an affine algebraic subset X.


Exercise I.1.1

(a) Let Y be the plane curve y = x^2. Show that A(Y) is isomorphic to a polynomial ring in one variable over k.

(b) Let Z be the plane curve xy = 1. Show that A(Z) is not isomorphic to a polynomial ring in one variable over k.

(c) Let f be any irreducible quadratic polynomial in k[x, y], and let W be the conic defined by f. Show that A(W) is isomorphic to A(Y) or A(Z). Which one is it when?