Example of a Dedekind Domain which is not a PID

Question

I am asked to show that $\mathbb R[X,Y]/(X^2+Y^2-1)$ is a DD but not a PID.

Some quick observations I made are it is Noetherian, Normal (since $X^2-1$ is square free).

How do I show the following two claims?

1. All non zero prime ideals are maximal
2. Is is NOT a PID.

Answer 1

Hint:

1. For any field $F$, $F[X,Y]$ has Krull dimension $2$ and $X^2+Y^2-1$ generates a prime ideal in $\mathbf R[X,Y]$ since it is irreducible and the polynomial ring is a U.F.D.