TITLE: Show these polynomial rings are not isomorphic
QUESTION [8 upvotes]: I wish to show $\mathbb{R}[X,Y]/(X^2 + Y^2 -1)$ is not isomorphic to $\mathbb{R}[T,T^{-1}]$ (where $X,Y,T$ are formal variables in the bigger polynomial rings$. 
I have tried to compare the basic properties, e.g. they are both integral domains but they seem to agree. Can I exploit the way $\mathbb{R}$ is not algebraically closed somehow? Since I have shown the result is true for $\mathbb{C}$ in place of $\mathbb{R}$.

REPLY [9 votes]: $\mathbb{R}[T, T^{-1}]$ is the localization of a UFD (a PID, even), hence is a UFD.  However, $\mathbb{R}[X,Y]/(X^2 + Y^2 - 1)$ is not a UFD: the idea is that $X^2 = (1-Y)(1+Y)$ gives two different factorizations.  For details, see this answer.
The interesting part is that if we work over $\mathbb{C}$, the two corresponding rings $\mathbb{C}[X,Y]/(X^2 + Y^2 - 1)$ and $\mathbb{C}[T,T^{-1}]$ are isomorphic:
$$
\mathbb{C}[T,T^{-1}] \cong \mathbb{C}[T,U]/(TU - 1) \cong \mathbb{C}[X,Y]/(X^2 + Y^2 - 1)
$$
where the last isomorphism sends $T \mapsto X + iY$ and $U \mapsto X - iY$.