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  "body": "\n## Statement\n\nLet $k$ be an [[algebraically closed field|02SJG1HC32]] of [[characteristic|1ZB13N78XW]] $0$.\nLet $P$ be the field of [[Puiseux series|SENTYNSKSF]] in $X$ over $k$.\nThen $P$ is algebraically closed, and it is an [[algebraic closure|0MCXX29HWY]] of the [[field of formal Laurent series|YZWYF7DXTW]] $k((X))$.\n\nIn particular, every polynomial in $Y$ of degree $n \\ge 1$ with coefficients in $k((X))$ has $n$ roots in $P$, counted with multiplicity.\n\n## Remarks\n\nProof idea (Newton's method): let $F(Y)$ be a polynomial with coefficients in $P$.\nA slope of the [[Newton polygon|H7QNCE2QZJ]] of $F$ for the $X$-adic valuation gives a possible leading exponent $\\gamma$ of a root.\nA root $c \\ne 0$ in $k$ of an associated polynomial gives the leading term $c X^\\gamma$.\nOne then substitutes $Y = X^\\gamma(c + Y_1)$ and repeats.\nIn characteristic $0$ one shows that the denominators of the exponents stop growing, so the process produces a Puiseux series.\nA second proof shows that every finite extension of $k((X))$ of degree $n$ is $k((X^{1/n}))$.\nExample: the roots of $Y^2 - X^3$ are $\\pm X^{3/2}$, and the roots of $Y^2 - X^2(1 + X)$ are $\\pm X(1 + X)^{1/2}$, with $(1 + X)^{1/2}$ given by the binomial series.\nThe characteristic $0$ hypothesis is necessary: over an algebraic closure of $\\F_p$, the polynomial $Y^p - Y - X^{-1}$ has no root in the field of Puiseux series.\n\n## Explanations\n\n### L1 human\n\nSolving a polynomial equation in $Y$ whose coefficients are power series in $X$ usually needs fractional powers of $X$.\nThe theorem says that fractional powers with a common denominator are always enough, when the coefficients come from an algebraically closed field such as $\\C$.\nFor example $Y^2 = X^3$ is solved by $Y = X^{3/2}$.\n\n### L2 human\n\nGeometrically, the theorem parametrizes each branch of a plane algebraic curve $F(X, Y) = 0$ through a point by $X = t^m$, $Y = $ a power series in $t$.\nThe exponents of the Puiseux expansion are the start of the classification of plane curve singularities.\nOver $\\C$ the Puiseux series that arise from polynomials converge near $0$, and they describe the branches analytically.\nThe Newton polygon is the computational tool, and [[Henselian fields|3ACEE4N6NE]] give the general frame for lifting approximate roots.\n",
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