irreducible polynomial

**Orbis** · April 4th 2010, 04:26 PM

Hi guys,

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

Thanks in advance,
Orbis

**chiph588@** · April 4th 2010, 04:43 PM

Originally Posted by Orbis

Hi guys,

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

Thanks in advance,
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 $2$ and $1$ , since $-2=1$ and $-1=2$ .

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

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

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

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

**Orbis** · April 4th 2010, 05:00 PM

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

Thanks for the quick reply!

Orbis

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