Gauss's Lemma

Introduction

Gauss's Lemma is a cornerstone result in ring theory that establishes the relationship between factorization in a ring and factorization in its polynomial ring. It is fundamental to understanding unique factorization in polynomial rings.

Statement

Theorem 11.2 (Gauss's Lemma and its Corollary): If $R$ is a UFD, then the polynomial ring $R[x]$ is also a UFD.

Key Concepts

Content of a Polynomial

The content of a polynomial $f(x)=a_n{x}^n+\cdots +a_{1}x+a_0$ in $R[x]$ is the greatest common divisor of its coefficients: $\text{cont}(f)=gcd(a_n,\dots ,a_0)$.

Primitive Polynomial

A polynomial $f(x)\in R[x]$ is primitive if its content is a unit in $R$, i.e., $\text{cont}(f)=1$.

Gauss's Lemma

Gauss's Lemma: The product of two primitive polynomials is primitive.

Proof Sketch

The proof involves:

1. Assuming that the product of two primitive polynomials is not primitive
2. Using the fact that $R$ is a UFD to find a prime element that divides all coefficients of the product
3. Deriving a contradiction by showing that this prime must divide all coefficients of one of the original polynomials

Corollary: UFD Property

Using Gauss's Lemma, one can prove that if $R$ is a UFD, then $R[x]$ is also a UFD.

Proof Strategy

1. Define the content: For any polynomial $f(x)\in R[x]$, write $f(x)=\text{cont}(f)\cdot f_1(x)$ where $f_1(x)$ is primitive.

2. Relate to field of fractions: Consider factorization in $F[x]$, where $F$ is the field of fractions of $R$.

3. Lift factorization: Use Gauss's Lemma to lift the unique factorization from $F[x]$ back to $R[x]$.

Consequences

This theorem is immensely powerful as it provides a vast supply of UFDs. For example:

• Since $\mathbb{Z}$ is a UFD, so is $\mathbb{Z}[x]$
• By induction, the polynomial ring in any number of variables over a UFD, such as $\mathbb{Z}[x_1,\dots ,x_n]$ or $F[x_1,\dots ,x_n]$ for a field $F$, is also a UFD

Examples

Example 1: Polynomial Rings over Fields

For any field $F$, the polynomial ring $F[x]$ is a UFD. This follows from the fact that $F$ is a UFD and applying Gauss's Lemma.

Example 2: Multivariable Polynomial Rings

The polynomial ring $\mathbb{Z}[x,y]$ is a UFD, even though it is not a PID. The ideal $(x,y)$ is not principal.

Example 3: Polynomial Rings over UFDs

If $R$ is any UFD, then $R[x_1,\dots ,x_n]$ is also a UFD for any number of variables.

Applications

Application 1: Polynomial Factorization

Gauss's Lemma is fundamental to understanding polynomial factorization over different rings.

Application 2: Algebraic Number Theory

The result is crucial in algebraic number theory for understanding factorization in rings of integers.

Application 3: Algebraic Geometry

Polynomial rings over UFDs are important in algebraic geometry for studying coordinate rings of varieties.

Related Concepts

• Unique Factorization Domains
• Polynomial Rings
• Content of Polynomials
• Primitive Polynomials
• Field of Fractions