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Let $R$ be the ring of functions that are polynomials in $\cos t$ and $\sin t$ with real coefficients.

  1. Prove that $R$ is isomorphic to $\mathbb R[x,y]/(x^2+y^2-1)$.

  2. Prove that $R$ is not a unique factorization domain.

  3. Prove that $S=\mathbb C[x,y]/(x^2+y^2-1)$ is a principal ideal domain and hence a unique factorization domain.

  4. Determine the units of the rings $S$ and $R$. (Hint: Show that $S$ is isomorphic to the Laurent polynomial ring $\mathbb C[u,u^{-1}]$.)

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    if you find an answer to your question helpful, you should consider accept it. See http://meta.math.stackexchange.com/questions/3286/how-do-i-accept-an-answer2012-12-13
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    @QiL'8 It's verbatim from Michael Artin's Algebra without explicating any effort or explanation, therefore downvote.2013-07-28

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