Trigonometric polynomial rings and their factorization properties

Abstract

Consider the rings $S$ and $S^{\prime }$, of real and complex trigonometricpolynomials over the field ${Q}$ and its algebraic extension${Q}(i)$ respectively. Then $S$ is an FFD, whereas $S^{\prime}$is a Euclidean domain. We discuss irreducible elements of $S$ and$S^{\prime}$, and prove a few results on the trigonometric polynomialrings $T$ and $T^{\prime}$ introduced by G. Picavet and M. Picavet in [Trigonometric 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 polynomialsin terms of irreducibles (atoms), to study the construction of thesepolynomials from irreducibles, which gives a geometric view of this study.