6
$\begingroup$

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

$\endgroup$
  • $\begingroup$ 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
4
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.