Question

Let f be a field. Is f[x1,x2] a principal ideal domain?

Answer 1

Answer:

for example, (x,y) is a maximal ideal which is not principal in F[x,y].

And also, F[x,y]/(x2−y2) is not an integral domain since (x−y)(x+y)=x2−y2. On the other hand, the polynomial y2−x is irreducible and hence F[x,y]/(x−y2) is an integral domain.