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.
Best Answer
Let $K$ be any field of characteristic $p$ with more than $p$ elements. Then $x^p-x$ has only the $p$ elements of the prime field as roots. Pick $b$ not in the prime field and let $a=-(b^p-b)$. Then $x^p-x+a$ has $b$ as root (and $b+k$ for $k$ in the prime field).