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I was given an exercise to show $A=\frac{\mathbb{C}[X,Y]}{(X^2+Y^2-1)}$ is a PID. But I wonder if it is true. Note that PID $\implies$ UFD. But we have $$X\cdot X = 1-Y^2 =(1-Y)(1+Y)$$ in $A$ which contradicts UFD.

Is there something wrong in the above factorization? Any suggestions / hints.

Edit: I can prove mechanically it is PID. My main concern was the above factorization. Thanks for the comments.

user26857
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Sayan
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    See https://mathoverflow.net/questions/5591/are-quotients-of-polynomial-rings-almost-ufds @ZachTeitler Here the base field is $\mathbb{C}$, which allows you to break $x^2+y^2=(x+iy)(x-iy)$. You will not have this decomposition when the base field is $\mathbb{R}$ or $\mathbb{Q}$. – Krish Aug 31 '17 at 06:11
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    This is a good question. I've seen many solutions of some similar questions based on such a quick "argument", but this isn't enough. You have to show that $x,1-y,1+y$ are irreducible and non-associates. But $x$ (for instance) is not irreducible in $A$ since we have $x=\frac{(x+iy+i)(x+iy-i)}{2(x+iy)}$. – user26857 Aug 31 '17 at 07:08
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    Oh yes, I see. And $1+y = \frac{1}{2}(x-iy)(x+iy+i)^2$. So it's not that $x$ is associated to $1\pm y$, it's that $x$ and $1\pm y$ are reducible. – Zach Teitler Aug 31 '17 at 14:46

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The answer is essentially here, but I will detail it to get this off the unanswered questions list.

As Georges Elencwajg observed in his answer to the corresponding MO post, the linear substitution $u:=x+iy$ and $v:=x-iy$ is invertible and therefore $\Bbb C[u,v]=\Bbb C[x,y]$. Furthermore, $$ uv = (x+iy)(x-iy)= x^2+y^2 $$ so the ring in question is $\Bbb C[u,v]/(uv-1)$, which is the same as $\Bbb C[u,u^ {-1}]$. Now, let us invert this linear transformation to understand the factorization of $x^2$.

We have $x=u-iy$ and $y=i(v-x)$. Hence, $x=u+v-x$, meaning $$ x=\frac{u+v}2=\frac{u+u^{-1}}2=\frac{u^2+1}{2u}=\frac{u-i}{2u}\cdot\frac{u+i}{2u} $$ is not even irreducible and similarly, $$ y = i\cdot \left(v-\frac{u+v}2\right)=i\cdot\frac{v-u}2=\frac{u-u^{-1}}{2i} = \frac{u^2-1}{2iu}=\frac{u-1}{2iu}\cdot\frac{u+1}{2iu} $$ isn't either. Now, $$ 1-y=\frac{2iu-u^2+1}{2iu}=\frac{(u-i)^2}{2ui} $$ and $$ 1+y = \frac{2iu+u^2-1}{2iu} = \frac{(u+i)^2}{2iu} $$ explains that you have simply redistributed your irreducible factors and it all really makes sense.

Jesko Hüttenhain
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