The ring of real trigonometric polynomials

The ringThroughout we consider the polynomial ringR=\mathbb{R}[\cos{x},\sin{x}].This ring has a lot of non-trivial properties which give us a good chance to study commutative ring theory. First of all note it is immediate thatR \cong \mathbb{R}[X,Y]/(X^2+Y^2-1)if the map is given by $X \mapsto \cos x$ and $Y \mapsto \sin x$. Besides, in $R$ we have\sin^2x=(1-\cos{x})(1+\cos{x})=\sin{x}\cdot\sin{x}which is to say that $R$ is not a factorial ring, although $\mathbb{R}[X,Y]$ is.This blog post is in...

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