Ring of trigonometric polynomials

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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.