What is the best way to understand that $D:=\mathbb{Q}[X,Y]/(X^2+Y^2-1)$ is a Dedekind domain?

I first noticed that $X^2+Y^2-1$ is irreducible in $\mathbb{Q}[X,Y]$ since it is $Y-1$ Eisenstein in $\mathbb{Q}[Y][X]$. It follows that $\mathbb{Q}[X,Y]/(X^2+Y^2-1)$ is an integral domain. By Hilbert Basis theorem, $\mathbb{Q}[X,Y]$ is Noetherian, so this quotient is too.

I know that a Dedekind domain is precisely a Noetherian integral domain which is integrally closed in its fraction field, and has Krull dimension $1$. However, computing the fraction field and showing $D$ is integrally closed in it seems quite difficult, and showing the Krull dimension is 1 also seems difficult.

I'm aware of another result that a Noetherian integral domain is Dedekind domain if the localization at every prime is a discrete valuation ring. I think the prime ideals of $D$ are precisely the canonical images of the prime ideals containing $(X^2+Y^2-1)$ in $D$. But I'm stuck trying to get a general handle on $D_P$ and seeing it is a DVR. What is the best way to see this claim (**preferably algebraically, not geometrically**)? Thanks.