Show tht $f(x)$ is either irreducible or it splits in $F$.

Question

Let $F$ be a field of characteristic $p$ and let $f(x)=x^p-a\in F[x].$ Show that $f(x)$ is either irreducible or it splits in $F$.

Attempt:

If $b=a^{\frac{1}{p}}\in F$ then $x^p-a=x^p-(a^{\frac{1}{p}})^p=(x-b)^p$. Thus $f$ splits in $F$.

Now if $b\notin F$, I need to show that $f$ is irreducible. I am unable to understand how to do that .

Will someone please help me to complete this?

Answer 1

In general $b$ lies in some extension $K/F$. Also $b^p\in F$. If $(X^p-a)=(X-b)^p$ is reducible over $F$ then each irreducible factor over $K$ has the form $(X-b)^r$ where $1\le r<p$. The $X^{r-1}$ coefficient of this is $-rb$ which must be in $F$....