1. ## irreducible

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

2. 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 $n=p^k, \ k \geq 0.$

3. 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 $n=p^k, \ k \geq 0.$
Why does $x^{p^{p^k}} - x + m^{-1}$ divide $x^{p^n}-x+1.$?
Please...give me some more detail explanation.