Is the hyperbola isomorphic to the circle?

Question

Is the ring $B=\mathbb{C}[x,y]/(xy-1)$ isomorphic with $C=\mathbb{C}[x,y]/(x^2+y^2-1)$?

I think they shouldn't but all my tryings fail to prove the fact. Are they in fact isomorphic so I may try to prove that. Thanks

Answer 1

Yes, they are isomorphic.

If $u = x+yi$, and $v = x-yi$ then $$\mathbb{C}[x,y]/(x^2+y^2-1)\simeq\mathbb C[u,v]/(uv-1).$$

Answer 2

This can be viewed as a question in algebraic geometry but also as a question in projective geometry. Both the hyperbola and the circle are conic sections, and are projectively equivalent. In homogeneous coordinates this follows from the fact that any pair of nondegenerate indefinite quadratic forms are equivalent.

Answer 3

For fun:

Let $\ \phi : \mathbb C \rightarrow \mathbb C\ $ be an arbitrary bijection (doesn't have to be continuous or measurable or anything, just a set-theoretical bijection). Let $\ A\,\ B\subseteq \mathbb C^\mathbb C\ $ be such that $\ B=\{f\circ\phi: f\in A\}.\ $ Then the rings $\ \mathbb C[A]\,\ \mathbb C[B]\ $ are isomorphic (as subrings of $\ \mathbb C^\mathbb C).\ $

Thus, rings $\ \mathbb C[ch\,\ i\!\cdot\! sh]\ $ and $\ \mathbb C[cos\,\ sin]\ $ are isomorphic via bijection $\ z\ \mapsto\ i\cdot z;\ $ and these two are respectively isomorphic to $\ \mathbb C[ch\,\ sh]\ $ and $\ \mathbb C[cos\,\ sin],\ $ and these last two are respectively isomorphic to the rings from the OP's post.