Ring of trigonometric polynomials

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This is a variation of polynomial ring
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Definition

Let R be a commutative unital ring. The ring of trigonometric polynomials or ring of circular polynomials over R is defined as the ring R[x,y]/(x^2 + y^2 - 1). Equivalently, it is the ring R[\cos t, \sin t] where \cos t and \sin t are subject to the usual relation \cos^2 t + \sin^2 t = 1.

Ring properties

Affine ring

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

Thus, the ring R[x,y]/(x^2 + y^2 - 1) is a Noetherian ring whenever R is a Noetherian ring.

Integral domain

When R has characteristic two, we have x^2 + y^2 - 1 = (x + y + 1)^2, so the quotient is the ring R[x,y]/(x+y+1)^2 which is a local ring with unique maximal ideal generated by x + y + 1.

On the other hand, when R is a field of characteristic not equal to two, the polynomial x^2 + y^2 -1 is irreducible over R[x,y]. Thus, the quotient R[x,y]/(x^2 + y^2 - 1). For full proof, refer: Ring of trigonometric polynomials over field of characteristic not equal to two is integral domain

Unique factorization domain

For R a field of characteristic not equal to two, the ring of trigonometric polynomials over R is a unique factorization domain if and only if -1 is a square in the field.