# Gauss's Lemma (Polynomial Theory)

## Contents

## Theorem

Gauss's lemma on polynomials may refer to any of the following statements.

## Product of primitive polynomials is primitive

### Rational Polynomial

Let $\Q$ be the field of rational numbers.

Let $\Q \left[{X}\right]$ be the ring of polynomials over $\Q$ in one indeterminate $X$.

Let $f \left({X}\right), g \left({X}\right) \in \Q \left[{X}\right]$ be primitive polynomials.

Then their product $fg$ is also a primitive polynomial.

### General Ring

Let $R$ be a commutative ring with unity.

Let $f,g \in R[X]$ be primitive polynomials.

Then $fg$ is primitive.

## Content is multiplicative

### Rational polynomials

Let $h \in \Q \left[{X}\right]$ be a polynomial with rational coefficients.

Let $c_h = \operatorname{cont} \left({h}\right)$ denote the content of $h$.

Then for any polynomials $f, g \in \Q \left[{X}\right]$ with rational coefficients:

- $\operatorname{cont} \left({f g}\right) = \operatorname{cont} \left({f}\right) \operatorname{cont} \left({g}\right)$

### Dedekind domain

Let $R$ be a Dedekind domain.

Let $f, g \in R[X]$ be polynomials.

Let $\operatorname{cont} (f)$ denote the content of $f$.

Then $\operatorname{cont} (fg) = \operatorname{cont} (f) \operatorname{cont} (g)$ is the product of $\operatorname{cont}(f)$ and $\operatorname{cont}(g)$.

## Statement on irreducible polynomials

Let $\Z$ be the ring of integers.

Let $\Z \left[{X}\right]$ be the ring of polynomials over $\Z$.

Let $h \in \Z \left[{X}\right]$ be a polynomial.

The following are equivalent:

- $(1):\quad$ $h$ is irreducible in $\Q \left[{X}\right]$ and primitive
- $(2):\quad$ $h$ is irreducible in $\Z \left[{X}\right]$.

## Polynomial ring is UFD

Let $R$ be a unique factorization domain.

Then the ring of polynomials $R \left[{X}\right]$ is also a unique factorization domain.

## Source of Name

This entry was named for Carl Friedrich Gauss.