# Polynomials in Z modulus 9

1. Aug 10, 2008

### 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 [3] and [6]). This basically is just showing that the only possible irreducible polynomials in $\mathbb{Z}_9$ are the constant ones, [3] and [6], and that all the other constant polynomials can be written as the product of polynomials with degrees greater than 0, kind of like how [1] can be written as,

$$([3]x+[1])([6]x+[1])=[0]x^2+[3]x+[6]x+[1]=[1]$$

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.