Prove that the quotient ring $\mathbb{C}[x,y]/(x^2+y^2-1)$ is a unique factorization domain.

I am trying to prove first it is a principal ideal domain. However I am really stuck on this problem

  • $\begingroup$ Have you tried writing $x=\cos x$, $y=\sin x$? It's only a suggestion, I don't know if it is the right way. $\endgroup$ – ajotatxe Apr 6 '14 at 18:35
  • $\begingroup$ @ajotatxe: I had not tried that but I am trying to use that substitution and still got nowhere. I am trying to use the idea of adjoining elements for this one $\endgroup$ – Rutherford Mark Apr 6 '14 at 19:08
  • 1
    $\begingroup$ Solved here. $\endgroup$ – user26857 Apr 10 '14 at 21:07

Show that the ring is isomorphic to the ring of complex trigonometric polynomials $\mathbb{C}[e^{i\theta},e^{-i\theta}]$. This is a localization of $\mathbb{C}[t]$ so is a PID.


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