By a result of Klein-Nagata rings of the form $A\_Q=K[x\_1,...,x\_n]/(Q)$ are factorial when $K$ is a field, $n \geq 5$ and $Q$ is a non-degenerate quadratic form.



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> Question 1: When is $A\_Q$ a principal ideal domain or even an euclidean ring for $n \geq 2$ for a general quadratic polynomial $Q$?
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> Question 2: What is the global dimension of $A\_Q$ when $Q$ is a quadratic form?
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When are rings of the form $K[x_1,...,x_n]/(Q)$ principal ideal domains when $Q$ is quadratic?
PID's have Krull dimension $1$ (or $0$, if you call a field a PID); $A\_Q$ will have Krull dimension $n-1$. So the only option is $n=2$ (the case $n=1$ doesn't apply since $k[x]/x^2$ is not a domain).


However, the $n=2$ case will also not give a PID. The ideal $\langle x\_1, x\_2, \ldots, x\_n \rangle$ is not even locally principal (compute the dimension of the Zariski tangent space).


When I say the title of this question, I thought it would be asking about $k[x,y]/(ax^2+bxy+cy^2+dx+ey+f)$. The answer is that this is a PID if the conic $ax^2+bxy+cy^2+dx+ey+f=0$ is smooth$^\*$ and either


(1) $a x^2+b xy + c y^2$ has roots in $\mathbb{P}^1\_k$ or


(2) the conic $ax^2+bxy+cy^2+dx+ey+f=0$ has no roots in $k$.


For example, $\mathbb{R}[x,y]/\langle x^2-y \rangle$, $\mathbb{R}[x,y]/\langle xy-1 \rangle$ and $\mathbb{R}[x,y]/\langle x^2+y^2+1 \rangle$ are all PID's, but $\mathbb{R}[x,y]/\langle x^2+y^2-1 \rangle$ is not.


$^\*$ In characteristic $2$, I should say regular instead of smooth. For example, I think that $\mathbb{F}\_2(t,u)[x,y]/(t x^2+u y^2+1)$ is a PID. I'm not completely confident in this.


