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.