Field of formal Laurent series

Summary

The field of formal Laurent series $k((X))$ over a field $k$ consists of formal sums $\sum_{n \ge N} a_n X^n$, see the definition. Only finitely many negative powers of $X$ occur. It is the field of fractions of the ring $k[\![X]\!]$ of formal power series. With the order of the lowest term it is a complete field for a discrete valuation, the equal-characteristic analogue of the p-adic numbers. It is the formal counterpart of the Laurent series of complex analysis, without any condition of convergence.