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  "body": "\n## Summary\n\nThe 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|2S4K8FA8QY]].\nOnly finitely many negative powers of $X$ occur.\nIt is the [[field of fractions|0F2PTGX1A2]] of the ring $k[\\![X]\\!]$ of [[formal power series|033WGN00BF]].\nWith the order of the lowest term it is a complete field for a [[discrete valuation|A2KKZAV29W]], the equal-characteristic analogue of the [[p-adic numbers|BT6T702C28]].\nIt is the formal counterpart of the [[Laurent series|0GXMPDNYX3]] of complex analysis, without any condition of convergence.\n\n## Explanations\n\n### L1 human\n\nA formal Laurent series is like a power series that may start with a few negative powers, such as $X^{-2} + 3X^{-1} + 1 + X + X^2 + \\cdots$.\nOne adds them term by term and multiplies them like polynomials; each coefficient of a product is a finite sum because there are only finitely many negative powers.\nEvery nonzero series has an inverse: $1/(X - X^2) = X^{-1}(1 + X + X^2 + \\cdots)$.\nSo these series form a field.\nNo question of convergence arises: the series are only lists of coefficients.\n",
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