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..
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 $\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} $.