Let $E$ a field, $b\in{E-E^p}$ and $\operatorname{char}(K)=p$, prove that $f(x)=x^{p^n}-b$ is irreducible.
I know that if $b\in{E-E^p}$ then there is no $a\in{E}$ such that $b=a^p$, also $x^p-b$ is irreducible in $E[x]$.
I also know that there is an $\alpha$ in some extension of $E$ such that $\alpha^p=b$, but I do not know if this helps me solve the problem.
I would appreciate your help with this exercise
Let $K$ be a field extension of $E$ where $b$ has a $p^n$th root $a$. In $K$, $X^{p^n} - b = (X-a)^{p^n}$
If $X^{p^n}-b = FG$, $F,G\in E[X]$, then $F=(X-a)^i, G = (X-a)^j$, $i,j<p^n$, then for instance $i=sp^k$ with $k<n$ and $s\land p = 1$.
Then $F=(X^{p^k} - a^{p^k})^s$. Expand this polynomial and look at the coefficient of the monomial of degree $s-1$: it's $-sa^{p^k}$. But $F\in E[X]$, so $sap^k \in E$. But $s$ is an integer, so it's also in $E$; moreover, since $s\land p =1$, it's invertible in $E$; so that $a^{p^k} \in E$. But since $k<n$ and $a^{p^n} = b$, this contradicts $b\notin E^p$.
So $X^{p^n}-b$ is irreducible