12
$\begingroup$

I have a question, I think it concerns with field theory.

Why the polynomial $$x^{p^n}-x+1$$ is irreducible in ${\mathbb{F}_p}$ only when $n=1$ or $n=p=2$?

Thanks in advance. It bothers me for several days.

  • 1
    Also. When $n = 1,$ and since $x^p \equiv x \pmod p$ by [FLT](http://en.wikipedia.org/wiki/Fermat's_little_theorem), we have $x^{p^n} - x +1 \equiv 1 \pmod p.$2012-03-19
  • 1
    @JD x is not in $\mathbb F_p$. The polynomial $x^p-x$ is not zero, it just has roots in $\mathbb F_p$.2012-03-19
  • 0
    If $p$ does not divide $n$, then the polynomial is divisible by $x^p-x+1/n$. I don't immediately see why $x^{p^p}-x^p+1$ couldn't be irreducible, though.2012-03-19
  • 0
    Dear @David, why is the polynomial divisible by $x^p-x+1/n$ ?2012-03-19
  • 0
    Set $g=x^p-x+1/n$. Then $g^{p^k} = x^{p^{k+1}} - x^{p^k} +1/n$. So $g^{p^{n-1}} + g^{p^{n-2}}+\cdots+g^p+g=x^{p^n}-x+1$.2012-03-20

1 Answers 1