How to prove the ring of Laurent polynomials over a field is a principal ideal domain?

abstract-algebra ring-theory

Polynomial ring over a field $k$ is a PID. Notice that $S^{-1}k[x]=k[x,x^{-1}]$, where $S=\{x^i:i\in \mathbb N\}$. Now use the fact that localization of a PID is a PID.


1st HINT: Given an ideal $I$ of $F[x,x^{-1}]$, what can you say about $I\cap F[x]$? How is the information you obtain related to $I$?

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