1. ## irreducible polynomial

Hi guys,

I was wondering how I can prove that $\displaystyle x^4+x+2$ in the field $\displaystyle \mathbb{Z}_3[x]$ is irreducible?
I know how to do it if the degree is 3 or less, but here I have no idea..

Orbis

2. Originally Posted by Orbis
Hi guys,

I was wondering how I can prove that $\displaystyle x^4+x+2$ in the field $\displaystyle \mathbb{Z}_3[x]$ is irreducible?
I know how to do it if the degree is 3 or less, but here I have no idea..

Orbis
It's easy to see there are no linear factors.

As for quadratic terms, WLOG its ok to assume the two constant terms are $\displaystyle 2$ and $\displaystyle 1$, since $\displaystyle -2=1$ and $\displaystyle -1=2$.

So we have $\displaystyle x^4+x+2=(x^2+ax+2)(x^2+bx+1) = x^4+(a+b)x^3+abx^2+(a+b)x+2$

The $\displaystyle x^3$ term tells us that $\displaystyle a+b\equiv0\bmod{3}$.

But the $\displaystyle x$ term tells us that $\displaystyle a+b\equiv1\bmod{3}$.

This is impossible, hence $\displaystyle x^4+x+2$ is irreducible in $\displaystyle \mathbb{Z}/3\mathbb{Z}$.

3. Ah, so simple! I was thinking along the lines of proving that $\displaystyle <x^4+x+2>$ was a maximal ideal, but that wasn't getting me anywhere.