# 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...