# irreducible

• Dec 9th 2008, 06:46 PM
Stiger
irreducible
Prove that \$\displaystyle x^{p^n}-x+1\$ is irreducible over \$\displaystyle \mathbb{F}_p\$ only when \$\displaystyle n=1\$ or \$\displaystyle n=p=2\$.
• Dec 9th 2008, 10:49 PM
NonCommAlg
Quote:

Originally Posted by Stiger
Prove that \$\displaystyle x^{p^n}-x+1\$ is irreducible over \$\displaystyle \mathbb{F}_p\$ only when \$\displaystyle n=1\$ or \$\displaystyle n=p=2\$.

if \$\displaystyle n=p^km,\$ where \$\displaystyle m > 1, \ k \geq 0, \ \gcd(p,m)=1,\$ then it's fairly easy to see that \$\displaystyle x^{p^{p^k}} - x + m^{-1}\$ will divide \$\displaystyle x^{p^n}-x+1.\$ so we may assume that \$\displaystyle n=p^k, \ k \geq 0.\$ (Wondering)
• Dec 11th 2008, 06:27 AM
Stiger
Quote:

Originally Posted by NonCommAlg
if \$\displaystyle n=p^km,\$ where \$\displaystyle m > 1, \ k \geq 0, \ \gcd(p,m)=1,\$ then it's fairly easy to see that \$\displaystyle x^{p^{p^k}} - x + m^{-1}\$ will divide \$\displaystyle x^{p^n}-x+1.\$ so we may assume that \$\displaystyle n=p^k, \ k \geq 0.\$ (Wondering)

Why does \$\displaystyle x^{p^{p^k}} - x + m^{-1}\$ divide \$\displaystyle x^{p^n}-x+1.\$?(Crying)
Please...give me some more detail explanation.