# Ring of trigonometric polynomials

This is a variation of polynomial ring

View a complete list of variations of polynomial ring OR read a survey article on varying polynomial ring

## Contents

## Definition

Let be a commutative unital ring. The **ring of trigonometric polynomials** or **ring of circular polynomials** over is defined as the ring . Equivalently, it is the ring where and are subject to the usual relation .

## Ring properties

### Affine ring

The ring of trigonometric polynomials is an affine ring over . In particular, when is a field, the ring of trigonometric polynomials is an affine ring over a field.

Thus, the ring is a Noetherian ring whenever is a Noetherian ring.

### Integral domain

When has characteristic two, we have , so the quotient is the ring which is a local ring with unique maximal ideal generated by .

On the other hand, when is a field of characteristic not equal to two, the polynomial is irreducible over . Thus, the quotient . *For full proof, refer: Ring of trigonometric polynomials over field of characteristic not equal to two is integral domain*

### Unique factorization domain

For a field of characteristic not equal to two, the ring of trigonometric polynomials over is a unique factorization domain if and only if is a square in the field.