# Simple extension is splitting field of $f=X^p-X+a \in K[X]$ with $char(K)=p$

Given is the polynomial $f=X^p-X+a \in K[X]$, where $K$ is a field of characteristic $p$ and $a \in K$.

My task now is to show that every simple extension of $f$ is already the splitting field of $f$.

I should already show, that if $f$ has no root in $K$, it is irreducible.

My professor gave the hint to make use of $f$'s seperability and the fact that $f(X)=f(X+1)$. I've already proven this by applying $f$ to $X+1$ and by using the criterium for seperabilty by which it is equivalent to show that $gcd(f,f')=1$, where $f'$ is the formal deviation of $f$.

• Hint: If $z$ is a zero of $f$, ANY zero, then $z+1$ is another. Apply that recursively. – Jyrki Lahtonen Dec 5 '17 at 14:30
• Really nice, thank you for this one. And if I did this $p$-times, I will end up on my "starting"-$z$ since $(...((z+1)+1)....+1)= z+p=z$ because $char(K)=p$. – Myrkuls JayKay Dec 5 '17 at 14:41
• Correct. Well done! – Jyrki Lahtonen Dec 5 '17 at 14:42