Possible Duplicate:
Ring of functions that are polynomials in $\cos t$ and $\sin t$, with real coefficients
Let $S=\mathbb R[X,Y]/(X^2+Y^2-1)$. Is the ring $S$ a UFD?
We obviously have a factorization $x^2=(1-y)(1+y)$ here but I have trouble showing that $x,1-y$ are irreducible, or that they are not equivalent (i.e. one is multiple of another by a unit in this ring). How would one go about proving these?
Is there an easier proof using the equivalence of UFD with Krull domain in which every height 1 prime ideal is principal?