The ring of trigonometric polynomials with complex scalars

In this post we have investigated some basic facts of \(\mathbb{R}[\cos{x},\sin{x}]\) (henceforth notations used there will be here as well). It is a half factorial domain with ideal class group \(\mathbb{Z}/2\mathbb{Z}\), and hence not a UFD. Next, we jump to the complex scalar field, and there are many nontrivial results. We will be discussing the ring\[R'=\mathbb{C}[\cos{x},\sin{x}].\]in a different style.Now we have a UFD (and an Euler's formula in disguise)Again, if we consider the map\[\be...

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