## You are here

HomeGauss's lemma I

## Primary tabs

# Gauss’s lemma I

There are a few different things that are sometimes called “Gauss’s Lemma”. See also Gauss’s Lemma II.

*Gauss’s Lemma I:* If $R$ is a UFD and $f(x)$ and $g(x)$ are both primitive polynomials in $R[x]$, so is $f(x)g(x)$.

*Proof:*
Suppose $f(x)g(x)$ not primitive. We will show either $f(x)$ or $g(x)$ isn’t as well. $f(x)g(x)$ not primitive means that there exists some non-unit $d$ in $R$ that divides all the coefficients of $f(x)g(x)$. Let $p$ be an irreducible factor of $d$, which exists and is a prime element because $R$ is a UFD. We consider the quotient ring of $R$ by the principal ideal $pR$ generated by $p$, which is a prime ideal since $p$ is a prime element. The canonical projection $R\to R/pR$ induces a surjective ring homomorphism $\theta:R[X]\to(R/pR)[X]$, whose kernel consists of all polynomials all of whose coefficients are divisible by $p$; these polynomials are therefore not primitive.

Since $pR$ is a prime ideal, $R/pR$ is an integral domain, so $(R/pR)[x]$ is also an integral domain. By hypothesis $\theta$ sends the product $f(x)g(x)$ to $0\in(R/pR)[X]$, which is therefore the product of $\theta(f(x))$ and $\theta(g(x))$, and one of these two factors in $(R/pR)[x]$ must be zero. But that means that $f(x)$ or $g(x)$ is in the kernel of $\theta$, and therefore not primitive.

## Mathematics Subject Classification

12E05*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff

## Recent Activity

new question: Prime numbers out of sequence by Rubens373

Oct 7

new question: Lorenz system by David Bankom

Oct 19

new correction: examples and OEIS sequences by fizzie

Oct 13

new correction: Define Galois correspondence by porton

Oct 7

new correction: Closure properties on languages: DCFL not closed under reversal by babou

new correction: DCFLs are not closed under reversal by petey

Oct 2

new correction: Many corrections by Smarandache

Sep 28

new question: how to contest an entry? by zorba

new question: simple question by parag