Consider the rings S and S , of real and complex trigonometric polynomials over the field Q and its algebraic extension Q(i) respectively. Then S is an FFD, whereas S is a Euclidean domain. We discuss irreducible elements of S and S , and prove a few results on the trigonometric polynomial rings T and T introduced by G. Picavet and M. Picavet in [Trigono- metric polynomial rings, Commutative ring theory, Lecture notes on Pure Appl. Math., Marcel Dekker, Vol. 231 (2003), 419–433]. We consider several examples and discuss the trigonometric polynomials in terms of irreducibles (atoms), to study the construction of these polynomials from irreducibles, which gives a geometric view of this study.
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