TITLE: $x^p-x+a$ irreducible for nonzero $a\in K$ a field of characteristic $p$ prime
QUESTION [8 upvotes]: Is it true that $f(x)=x^p-x+a\in K[x]$ is irreducible for nonzero $a\in K$ a field of characteristic $p$ prime?
I've seen variants of this question around, but they don't seem to answer the question as worded.  (It's possible I have not searched well enough or misunderstood the techniques already given)

I almost understand the case for finite fields: 
If $p=2$,  then to show irreducibility we need only show that it has no roots.  
For $p>2$, I can show that it's separable since the formal derivative is $-1$. Separability also follows from $f(\alpha)=0\Rightarrow f(\alpha+1)=0$.  This also shows that if $\alpha$ were a root in $\mathbb{F}_p$ then $0$ would be a root, a contradiction since we assumed $a\neq 0$; hence the polynomial has no roots in the prime subfield of $K$.  [...] But then?  Arguments I have seen seem to use the additional fact that $f(x)\in \mathbb{F}_p$.

I imagine it will come down to some kind of argument with coefficients (depending on roots, maybe using elementary symmetric polynomials) but other nifty ways I'm not seeing also appreciated.

REPLY [3 votes]: Or, just take $K=\mathbb Z_p[y]/\langle y^p-y+1\rangle$ for the most basic counterexample, since $y^p-y+1$ is prime in $\mathbb Z_p[y]$. Then But $x^p-x+1$ has a root $y$ in $K$.