Ring of trigonometric polynomials
This is a variation of polynomial ring
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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 .
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.
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.