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The ring of trigonometric polynomials with complex scalars

This post is a continuation of a previous post about the ring of trigonometric polynomials over the real field. Now we have jumped into the complex field, and the extension is not a trivial matter.

The ring of real trigonometric polynomials

The ring

Throughout we consider the polynomial ring

$$R=\mathbb{R}[\cos{x},\sin{x}].$$

This ring has a lot of non-trivial properties which give us a good chance to study commutative ring theory.