Vipul: New page: {{variation of|polynomial ring}} ==Definition== Let $R$ be a commutative unital ring. The '''ring of trigonometric polynomials''' or '''ring of circular polynomials''' ove...

2009-02-06T02:03:53Z

New page: {{variation of|polynomial ring}} ==Definition== Let <math>R</math> be a commutative unital ring. The '''ring of trigonometric polynomials''' or '''ring of circular polynomials''' ove...

New page

{{variation of|polynomial ring}}

==Definition==

Let <math>R</math> be a [[commutative unital ring]]. The '''ring of trigonometric polynomials''' or '''ring of circular polynomials''' over <math>R</math> is defined as the ring <math>R[x,y]/(x^2 + y^2 - 1)</math>. Equivalently, it is the ring <math>R[\cos t, \sin t]</math> where <math>\cos t</math> and <math>\sin t</math> are subject to the usual relation <math>\cos^2 t + \sin^2 t = 1</math>.

==Ring properties==

===Affine ring===

The ring of trigonometric polynomials is an affine ring over <math>R</math>. In particular, when <math>R</math> is a field, the ring of trigonometric polynomials is an [[affine ring over a field]].

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

===Integral domain===

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

On the other hand, when <math>R</math> is a field of characteristic not equal to two, the polynomial <math>x^2 + y^2 -1</math> is irreducible over <math>R[x,y]</math>. Thus, the quotient <math>R[x,y]/(x^2 + y^2 - 1)</math>. {{proofat|[[Ring of trigonometric polynomials over field of characteristic not equal to two is integral domain]]}}

===Unique factorization domain===

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

Ring of trigonometric polynomials - Revision history

Vipul: New page: {{variation of|polynomial ring}} ==Definition== Let R be a commutative unital ring. The '''ring of trigonometric polynomials''' or '''ring of circular polynomials''' ove...

Vipul: New page: {{variation of|polynomial ring}} ==Definition== Let $R$ be a commutative unital ring. The '''ring of trigonometric polynomials''' or '''ring of circular polynomials''' ove...