Is there an extension of a function field of degree $n$ for every $n$?

Question

Consider the field of formal Laurent series $\mathbb{C}((t))$, let $X$ be a curve defined over that field, and let $K$ be the field of functions of $X$. Given an integer $n \ge 2$, does there exist an extension of $K$ of degree $n$?

Also, what is a good reference for this material?

Answer 1

This field has negative powers of the variable $t$, but not fractional powers. So $\mathbf{C}((t^{1/n}))$ (I think this is called Puiseux series) should provide degree $n$ extension you are looking for.