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

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?

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.
• To my perception, the question literally is asking about curves over scalars $\mathbb C((t))$, so extension of the scalars is not quite to the point... ? Commented Jun 13, 2023 at 3:24
• Is it the case the function field of a curve over $\mathbb{C}((t))$ can be thought of as $\mathbb{C}((t))(s)$ for some indeterminate $s$, and then $\mathbb{C}((t^{1/n}))(s)$ is an extension of degree $n$? Commented Jun 13, 2023 at 4:09