irreducible

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

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

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

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

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