Let $F$ be a field, and $F[x,y]$ be a ring of polynomials in two variables. Is $F[x,y]$ a Principal Ideal Domain?
Also show that $F[x,y]/(y^2-x)$ and $F[x,y]/(y^2-x^2)$ are not isomorphic for any field $F$.
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No; for example, $(x,y)$ is a maximal ideal which is not principal in $F[x,y]$.
And also, $F[x,y]/(x^2-y^2)$ is not an integral domain since $(x-y)(x+y)=x^2-y^2$. On the other hand, the polynomial $y^2-x$ is irreducible and hence $F[x,y]/(x-y^2)$ is an integral domain.
If $A$ is a commutative ring, a classical result states that the polynomial ring $A[x]$ is a PID if and only if $A$ is a field. It is a good exercise.
In your case, as $F[x]$ isn't a field, $F[x,y] \simeq (F[x])[y]$ cannot be a PID. (I'm not claiming it's the best proof).
For the second question, Bruno's answer will be hard to improve upon.