[[K-monoid|$\mathbb K$-ring]]
# Algebra of Laurent polynomials

Let $\mathbb{K}$ be a field.
The algebra $\mathbb{K}[t, t^{-1}]$ of Laurent polynomials in indeterminate $t$ is a $\mathbb{Z}$-[[Graded algebra|graded]] commutative [[K-monoid|$\mathbb K$-ring]], with elements of the form
$$
\begin{align*}
f = \sum_{n \in \mathbb{Z}} f_{n}t^n
\end{align*}
$$
such that $f_{n}$ has finite [[Support of a map|support]],
with multiplication given by $t^n \cdot t^m = t^{n+m}$.
It is isomorphic to the [[group ring|group algebra]] $\mathbb{K}[\mathbb{Z}]$.

## Properties

1. The [[Degree operator|degree derivation]] is given formally by $d = t \frac{d}{dt}$ ^degreeDerivation
2. The derivations of $\mathbb{K}[t,t^{-1}]$ form the [[Witt algebra]].

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