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  "body": "\n## Definition\n\nLet $k$ be a [[field|0QA8CHH8MG]] and let $X$ be a variable.\nLet $m$ be a positive integer, $N$ an integer, and $a_n \\in k$ for all $n \\ge N$.\nA *Puiseux series* in $X$ over $k$ is a formal sum of the form\n$$f = \\sum_{n \\ge N} a_n X^{n/m}.$$\nEquivalently, $f$ is an element of the [[field of formal Laurent series|YZWYF7DXTW]] $k((X^{1/m}))$ in the variable $X^{1/m}$, for some $m \\ge 1$.\n\nFor $m \\mid m'$, the field $k((X^{1/m}))$ is identified with a subfield of $k((X^{1/m'}))$ through $X^{1/m} = (X^{1/m'})^{m'/m}$.\nThe field of Puiseux series is the union $\\bigcup_{m \\ge 1} k((X^{1/m}))$ with these identifications.\n\n## Remarks\n\nExample: $f = X^{1/2}$ is a Puiseux series with $m = 2$, and it is a root of $Y^2 - X$ in the field of Puiseux series, since $(X^{1/2})^2 = X$ in $k((X^{1/2}))$.\nNon-example: the formal sum $\\sum_{n \\ge 1} X^{1 - 1/n}$ has the exponents $0, 1/2, 2/3, 3/4, \\ldots$, whose denominators are unbounded, so there is no single $m$ and it is not a Puiseux series.\nThe union is a field because any two Puiseux series lie in a common field $k((X^{1/m}))$.\nThe order $v(f)$ of the lowest exponent with nonzero coefficient is a valuation on the field of Puiseux series with value group $\\Q$.\n",
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