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  "body": "\n## Statement\n\nLet $(A, \\mathfrak m)$ be a [[complete Noetherian local ring|G9Q61B8A7C]], and let $f = \\sum_{i \\ge 0} a_i X^i$ be an element of the [[formal power series|033WGN00BF]] ring $A[\\![X]\\!]$.\nSuppose that some coefficient $a_i$ does not lie in $\\mathfrak m$, and let $d$ be the least such index.\nThen there are a unique [[unit|0XVEZDE9J8]] $u$ of $A[\\![X]\\!]$ and a unique [[monic polynomial|0C3QBCR46M]]\n$$g = X^d + b_{d-1} X^{d-1} + \\cdots + b_1 X + b_0, \\qquad b_0, \\dots, b_{d-1} \\in \\mathfrak m,$$\nsuch that $f = u g$.\n\n## Remarks\n\nThe proof goes through the Weierstrass division theorem.\nUnder the same hypothesis, every $h \\in A[\\![X]\\!]$ can be written uniquely as $h = q f + r$ with $q \\in A[\\![X]\\!]$ and $r$ a polynomial of degree less than $d$.\nDividing $X^d$ by $f$ gives $X^d = q f + r$, and one checks that $q$ is a unit and $g = X^d - r$ has its lower coefficients in $\\mathfrak m$.\nThe division itself is built by successive approximation modulo powers of $\\mathfrak m$, which converges because $A$ is complete.\nThe hypothesis on the coefficients is needed.\nFor $A = k[\\![Y]\\!]$ and $f = Y$, every coefficient lies in $\\mathfrak m = (Y)$, and $f$ is not of the form $u g$.\nIndeed, modulo $\\mathfrak m$ the product $u g$ becomes $\\bar u X^d \\neq 0$, while $f$ becomes $0$.\nApplied to $A = k[\\![X_1, \\dots, X_{n-1}]\\!]$, the theorem says that a power series in $n$ variables which is nonzero modulo $(X_1, \\dots, X_{n-1})$ is, up to a unit, a polynomial in $X_n$.\nThe original form of the theorem is analytic.\nLet $f$ be a [[holomorphic function|0NAHXR1FRH]] near $0 \\in \\C^n$ such that $z_n \\mapsto f(0, \\dots, 0, z_n)$ has a zero of finite order $d$ at $0$.\nThen the same factorization holds with convergent series.\nHere $g$ is a monic polynomial in $z_n$ whose lower coefficients are holomorphic in $z_1, \\dots, z_{n-1}$ and vanish at $0$.\n\n## Explanations\n\n### L1 human\n\nA power series can have infinitely many terms, which makes it hard to handle.\nThe theorem says that, near the origin, such a series behaves like an ordinary polynomial in one of its variables, after multiplication by a series that never vanishes there.\nSo questions about zeros of power series reduce to questions about zeros of polynomials.\n\n### L2 human\n\nThe theorem is the main tool to prove, by induction on $n$, that $k[\\![X_1, \\dots, X_n]\\!]$ over a field $k$ is a Noetherian unique factorization domain.\nThe same holds for the ring of convergent power series in $n$ complex variables.\nIn number theory, for $A = \\Z_p$, it shows that a power series in $\\Z_p[\\![X]\\!]$ not divisible by $p$ is a unit times a polynomial, which is used in Iwasawa theory.\n",
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