$a \in \Bbb F_p $ but $a^{1/p} \not \in F_p$ proof that $f(x)=x^{p^n}-a $ is Irreducible over $F_p[x]$

Question

$a \in \Bbb F_p $ but $a^{1/p} \not \in F_p$ proof that $f(x)=x^{p^n}-a $ is Irreducible over $F_p[x]$

I know only the case when $n=1$ than if $\alpha$ is the root of the polynomial so $x^p-a= x^p-\alpha ^p =(x-\alpha)^p$ and so $\alpha$ is anti saperable root of the polynomial $f_\alpha$ its degree $deg(f_\alpha)\ge p $ but $ f_\alpha| x^p-a$ while $x^p-a$ is a fixed polynomial with a degree of $p$ so $f_\alpha (x)=x^{p}-a$ is Irredcible over $F_p(x)$

how I proof the case for $n > 1$?

Answer 1

Awfully sorry, but you’ve gone astray. For $a\in\Bbb F_p$, $a^p=a$, so that $X^p-a=(X-a)^p$ can not be irreducible in $\Bbb F_p[X]$.