Editing Laurent polynomial (section)

== Properties ==

* A Laurent polynomial over <math>\mathbb{C}</math> may be viewed as a [[Laurent series]] in which only finitely many coefficients are non-zero.
* The ring of Laurent polynomials <math>R\left [X, X^{-1} \right ]</math> is an extension of the [[polynomial ring]] <math>R[X]</math> obtained by "inverting <math>X</math>". More rigorously, it is the [[localization of a ring|localization]] of the polynomial ring in the [[multiplicative set]] consisting of the non-negative powers of <math>X</math>. Many properties of the Laurent polynomial ring follow from the general properties of localization.
* The ring of Laurent polynomials is a [[subring]] of the [[rational function]]s.
* The ring of Laurent polynomials over a field is [[Noetherian ring|Noetherian]] (but not [[Artinian ring|Artinian]]).
* If <math>R</math> is an [[integral domain]], the [[unit (ring theory)|units]] of the Laurent polynomial ring <math>R\left [X, X^{-1} \right ]</math> have the form <math>uX^{k}</math>, where <math>u</math> is a unit of <math>R</math> and <math>k</math> is an integer. In particular, if <math>K</math> is a field then the units of <math>K[X, X^{-1}]</math> have the form <math>aX^{k}</math>, where <math>a</math> is a non-zero element of <math>K</math>.
* The Laurent polynomial ring <math>R[X, X^{-1}]</math> is [[isomorphic]] to the [[group ring]] of the [[group (mathematics)|group]] <math>\mathbb{Z}</math> of [[Integer#Algebraic properties|integers]] over <math>R</math>. More generally, the Laurent polynomial ring in <math>n</math> variables is isomorphic to the group ring of the [[free abelian group]] of rank <math>n</math>. It follows that the Laurent polynomial ring can be endowed with a structure of a commutative, [[cocommutative]] [[Hopf algebra]].