A one dimensional field $K$ over a ground field $k$ contains $k[x]$ for $x \in K \setminus k$ such that it is a finitely generated $k[x]$-module.
The textbook I'm studying uses its geometric interpretation to motivate a study of Algebraic curve, but geometry aside for now, is it of any significance in Number theory or other fields? What are particular examples other than finite algebraic extensions of $k[x]$?