Exercise from Lang.
I am trying to show that if $char K = p$ and $a \in K$ has no $p^{th}$ root (implying that K is an infinite field) then $X^{p^n} - a$ is irreducible for all positive integer $n.$ I am stumped.
First I tried $n = 1.$ If $X^p - a$ is reducible then $\prod f_i = X^p - a$ where $1 < deg(f_i) < p$ and $f_i$ irreducible. Clearly $deg(f_i) > 1 $ as otherwise it would mean $a$ has a $p^{th}$ root in $K.$ Now $X^p - a$ is not separable by the first derivative test. Yet each $f_i$ is separable by the first derivative test. Hence, two of these polynomials have a common zero which is a contradiction.
I cannot, however, generalize this method for $n > 1.$ Is there another way to approach this?