# Polynomials in Z modulus 9

#### jeffreydk

I'm trying to figure out how to prove that every polynomial in $\mathbb{Z}_9$ can be written as the product of two polynomials of positive degree (except for the constant polynomials  and ). This basically is just showing that the only possible irreducible polynomials in $\mathbb{Z}_9$ are the constant ones,  and , and that all the other constant polynomials can be written as the product of polynomials with degrees greater than 0, kind of like how  can be written as,

$$(x+)(x+)=x^2+x+x+=$$

but I'm a bit lost on how to show it all, because it's not a field so the theorems I've been studying regarding irreducibility in polynomials don't apply to such a situation. Thanks, any help is greatly appreciated.

Related Linear and Abstract Algebra News on Phys.org

#### fresh_42

Mentor
2018 Award
If $f(x)\in \mathbb{Z}_9[x]$ is irreducible, and we have a decomposition $f(x)=f(x)\cdot 1=f(x)(3x+1)(6x+1)$ then either $f(x)(3x+1)\in \mathbb{Z}_9$ or $f(x)(6x+1)\in \mathbb{Z}_9$. In both cases we get $3f(x) =3a$ for an $a\in \mathbb{Z}_9\,.$ Thus $3(f(x)-a) = 0$ and $9\,|\,3(f(x)-a)$ which means $3\,|\,(f(x)-a)$ and $f(x)=3n+a$ for some $n\in \mathbb{Z}$. Hence $f(x)$ is constant, i.e. all irreducible polynomials are constant.

### Want to reply to this thread?

"Polynomials in Z modulus 9 "

### Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving