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I would like to know if the following is true.

Let $F$ be a field and let $p\in F[x]$ be a square-free polynomial. Then, the quotient ring $F[x,y]/\langle y^2-p\rangle$ is a UFD.

I am not sure how to approach this question.


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If you suspect it's not a UFD, you might look for a non-unique factorization. Something, perhaps, like $y^2 = p$... – Qiaochu Yuan Mar 6 '12 at 4:35
Yes, but I think it is a UFD. My problem is I do not know how to approach the problem. Thanks! – mississauga21 Mar 6 '12 at 20:12
What makes you think it's a UFD? – Qiaochu Yuan Mar 6 '12 at 20:16

No, it's not necessarily a UFD.

Here's an example, taken from Gauss's Lemma for Number Fields.

Let $F=\mathbb{C}$, and $p(x) = x^3-3x+49$. Then let $A=\mathbb{C}[x,y]/(y^2-p)$. $A$ is a Dedekind domain, so it is a UFD if and only if it is a PID.

(I'll be a bit sloppy below and write things like $y+7$ for elements of $A$, when in fact I mean $y+7+(y^2-p)$, the class of $y+7$.)

The ring $A$ is closely related to the elliptic curve $C: y^2=x^3-3x+49$; the nonzero prime ideals of $A$ are associated with points lying on the curve $C$ by the correspondence that sends $(a,b)$ to $(x-a,y-b)$.

The ideal $(x,7+y)$ is not principal in this ring (in fact, no power of $(x,7+y)$ is principal in the integral closure of $\mathbb{C}[x]$ in any finite extension of the ring of fractions of $A$). Therefore, $A$ is not a PID, hence not a UFD.

(The ideal $(x,7+y)$ corresponds to the point $(0,-7)$ on the curve; the ideal $(x,7+y)^k$ is principal if and only if the order of $(0,-7)$ in the group of points of $C$ divides $k$; since only countably many points on $C$ have finite order, "most" ideals will not be principal. In fact, the point $(0,-7)$ has infinite order, so the ideal $(x,7+y)^k$ is not principal for any $k$).

You can also verify that $A$ is not a UFD directly by noting that even though $x$ divides $(y+7)(y-7)$, and $y+7$ and $y-7$ are coprime, there is no way to factor $x$ in $A$ into a product $ab$ such that $a|y+7$ and $b|y-7$; if $A$ were a UFD, then this would be possible.

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