ABSTRACT
Let T (resp. T′) be the ring of real (resp. complex) trigonometric polynomials. Then T′ is a Euclidean domain while T is a half-factorial Dedekind domain. We characterize irreducible elements of T, the behavior of maximal ideals of T with respect to irreducible elements of T and the correspondence between maximal ideals of T and T′. Moreover, we consider the embedding of T into the ring of continuous numerical functions https://www.w3.org/1998/Math/MathML"> C https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429222641/bd60306a-e05b-46de-85c8-902aafc6d56a/content/eq3429.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> (ℝ) and give the link between prime ideals of T and https://www.w3.org/1998/Math/MathML"> C https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429222641/bd60306a-e05b-46de-85c8-902aafc6d56a/content/eq3430.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> (ℝ). As a by-product, we find that T can be equipped with total orders associated to ultrafilters on the field of real numbers.
