TITLE: Proof for Unique Factorization Domain
QUESTION [6 upvotes]: 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

REPLY [4 votes]: 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.