I have to show that $p(x)=x^5+5x+11$ is irreductible on $\mathbb{Z}[x]$ both using and not using Eisenstein. I have been trying to translate $p(x)$ or searching a field $\mathbb{Z}p$ but can't reach anywhere even with Eisenstein
Asked
Active
Viewed 65 times
2
-
Try to factor it $\pmod 3$. – lulu Jul 02 '18 at 15:07
-
Non-standard but nice criterion given by Osada (see this answer) applies here because $11$ is a prime and $11>1+5$. – Sil Jul 03 '18 at 18:49
1 Answers
6
You can use Eisensteins criterion on $p(x-1)$. Without using Eisenstein, you could note that $$p(x)\equiv x^5+2x+2\pmod{3},$$ which is irreducible as it has no roots in $\Bbb{F}_3$, and the three irreducible quadratics $x^2+x+2$, $x^2+2x+2$ and $x^2+1$ do not divide it.
Servaes
- 63,261
- 7
- 75
- 163