TITLE: Detecting whether some rings are Dedekind domains
QUESTION [9 upvotes]: 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.

REPLY [4 votes]: As said in the comments, $\mathbb{Z}[x,y]/(f)$ cannot have dimension $1$ so cannot be a Dedekind domain.
For any field $k$ and any nonzero polynomial $f(x,y) \in k[x,y]$, the quotient $k[x,y]/(f)$ is the coordinate ring of the plane algebraic curve $f(x,y) = 0$.  This ring is a domain iff $f$ is irreducible.  It is always be Noetherian and one-dimensional.  It is a Dedekind domain iff it is nonsingular (nonsingular $\implies$ normal $\implies$ nonsingular in codimension one).
Assuming that the ground field $k$ is perfect, nonsingular is equivalent to smooth, and the latter is much easier to detect.  The way to do it goes back essentially to multivariable calculus: the curve defined by $f(x,y) = 0$ is smooth at a point $P = (x,y)$ (such that $f(P) = 0$!) iff at least one of $\partial f / \partial x, \partial f / \partial y$ is nonzero at $P$.  
Thus for instance in your third example both partial derivatives vanish at $P = (0,0)$.  What about the first example?