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Prove $$\mathbb { R } \left[ x _ { 1 } , \ldots , x _ { n } \right] / \left( x _ { 1 } ^ { 2 } + \cdots + x _ { n } ^ { 2 } - 1 \right)$$

(i) is not a UFD when $n=2$

(ii) is a UFD when $n>2$

Part (i) is already solved, see Ring of trigonometric functions with real coefficients.

What about part (ii)? Is there any intuitive way to see why this is a UFD? A hint is more than enough. Thanks in advance.

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  • $\begingroup$ Hints. (I denote your ring by $R$) 1) $t:=1-x_n$ is prime in $R$; 2) $R_t=\mathbb R[t^{-1}x_1,\dots,t^{-1}x_{n-1},t]$; 3) $\mathbb R[t^{-1}x_1,\dots,t^{-1}x_{n-1}]$ is isomorphic to a polynomial ring over $\mathbb R$; 4) Apply Nagata criterion for factoriality. $\endgroup$ – user26857 Feb 26 at 15:50
  • $\begingroup$ See also this duplicate. $\endgroup$ – Dietrich Burde Feb 26 at 18:19

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