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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?

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    Dear Bernard, I have added another answer to the previous question on this which is more geometric, and possibly of interest (depending on your background). Regards,2012-12-17

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