# Formal Puiseux Series

Manuel Eberl

17 February 2021

### Abstract

Formal Puiseux series are generalisations of formal power series and formal Laurent series that also allow for fractional exponents. They have the following general form: $\sum_{i=N}^\infty a_{i/d} X^{i/d}$ where N is an integer and d is a positive integer.

This entry defines these series including their basic algebraic properties. Furthermore, it proves the Newton–Puiseux Theorem, namely that the Puiseux series over an algebraically closed field of characteristic 0 are also algebraically closed.