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  "body": "\n## Definition\n\nLet $k$ be a [[field|0QA8CHH8MG]] and let $X$ be a symbol.\nA *formal Laurent series* in $X$ over $k$ is a family $(a_n)_{n \\in \\Z}$ of elements of $k$ with the following property.\nThere is $N \\in \\Z$ with $a_n = 0$ for all $n \u003c N$.\nIt is written $\\sum_{n \\ge N} a_n X^n$.\nThe set $k((X))$ of these series has the operations\n$$\\sum_n a_n X^n + \\sum_n b_n X^n = \\sum_n (a_n + b_n) X^n, \\qquad \\Big(\\sum_n a_n X^n\\Big)\\Big(\\sum_n b_n X^n\\Big) = \\sum_n \\Big(\\sum_{i + j = n} a_i b_j\\Big) X^n.$$\nWith these operations $k((X))$ is a field, the *field of formal Laurent series* over $k$.\n\n## Remarks\n\nThe inner sum $\\sum_{i + j = n} a_i b_j$ is finite: if $a_i = 0$ for $i \u003c N$ and $b_j = 0$ for $j \u003c M$, only the pairs with $N \\le i \\le n - M$ contribute.\nExample: $f = X - X^2$ has the inverse $g = \\sum_{n \\ge -1} X^n = X^{-1} + 1 + X + \\cdots$; indeed $f g = (1 - X)(1 + X + X^2 + \\cdots) = 1$.\nNon-example: the family with $a_n = 1$ for all $n \\in \\Z$ is not a formal Laurent series, since no $N$ bounds its negative powers.\nIts square would need the infinite sum $\\sum_{i + j = 0} 1$.\nThe map $v(f) = \\min \\set{n : a_n \\ne 0}$ for $f \\ne 0$ is a [[discrete valuation|ZYK5F0GBK6]] on $k((X))$ with uniformizer $X$, and $k((X))$ is complete for it.\nIts valuation ring is the ring $k[\\![X]\\!]$ of [[formal power series|072W67KFYK]], and $k((X))$ is the [[field of fractions|0VF11ZTTMD]] of $k[\\![X]\\!]$.\n",
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