Consider the three rings $\mathbb{C}[x,y] / \langle x^4 + xy -1\rangle$, $\mathbb{Z}[x,y] / \langle x^4 + xy -1\rangle$ and $\mathbb{F}_2[x,y] /\langle x^4- y^3 \rangle$. I am supposed to detect whether these are Dedekind domains or not.
However I've no idea how to do this. I know that a Dedekind domain is normal, noetherian, and of dimension $1$.
So I can at least see that each of these is noetherian, because they are quotients of noetherian rings. But I have no idea how to verify the other things, or unverify them. Are there any standard methods or tricks to work this out? I would really appreciate any help on this, thank you.