---
tag: 49JSD49P7X
kind: statement
role: theorem
name: Weierstrass preparation theorem
msc: [13J05, 32B05, 13F25]
uses: [G9Q61B8A7C, 033WGN00BF, 0XVEZDE9J8, 0C3QBCR46M, 0NAHXR1FRH]
about: [033WGN00BF]
sources:
  - url: https://en.wikipedia.org/wiki/Weierstrass_preparation_theorem
same_as:
  wikidata: Q3527219
  wikipedia: Weierstrass preparation theorem
---

## Statement

Let $(A, \mathfrak m)$ 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 $A[\![X]\!]$.
Suppose that some coefficient $a_i$ does not lie in $\mathfrak m$, and let $d$ be the least such index.
Then there are a unique [[unit|0XVEZDE9J8]] $u$ of $A[\![X]\!]$ and a unique [[monic polynomial|0C3QBCR46M]]
$$g = X^d + b_{d-1} X^{d-1} + \cdots + b_1 X + b_0, \qquad b_0, \dots, b_{d-1} \in \mathfrak m,$$
such that $f = u g$.

## Remarks

The proof goes through the Weierstrass division theorem.
Under 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 $d$.
Dividing $X^d$ 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 $\mathfrak m$.
The division itself is built by successive approximation modulo powers of $\mathfrak m$, which converges because $A$ is complete.
The hypothesis on the coefficients is needed.
For $A = k[\![Y]\!]$ and $f = Y$, every coefficient lies in $\mathfrak m = (Y)$, and $f$ is not of the form $u g$.
Indeed, modulo $\mathfrak m$ the product $u g$ becomes $\bar u X^d \neq 0$, while $f$ becomes $0$.
Applied 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 $X_n$.
The original form of the theorem is analytic.
Let $f$ 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 $0$.
Then the same factorization holds with convergent series.
Here $g$ is a monic polynomial in $z_n$ whose lower coefficients are holomorphic in $z_1, \dots, z_{n-1}$ and vanish at $0$.

## Explanations

### L1 human

A power series can have infinitely many terms, which makes it hard to handle.
The 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.
So questions about zeros of power series reduce to questions about zeros of polynomials.

### L2 human

The 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.
The same holds for the ring of convergent power series in $n$ complex variables.
In 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.