I want to show that $f(x)=x^p - x+ a$ is irreducible in $\mathbb{F}_p [x]$ if $a \neq 0$. I know that if $b$ is a root of $f$, then $b+1$ is also a root of $f$. Can I use this fact to prove that $f$ is irreducible? Any hints or reference are appreciated.
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That's a good start. But this question has been asked many times. In the dupe target there are several nice ways of getting there (mine is among the least thrilling). – Jyrki Lahtonen Nov 03 '18 at 10:16
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We know $f$ has no roots in $\mathbb{F}_p$.
Suppose $g(x)\in\mathbb{F}_p[x]$ is a degree $d>1$ irreducible factor of $f(x)$. Then $g(x+m)$ for all $m\in\mathbb{F}_p$ are irreducible factors of $f(x)$. They can't be all distinct because that would be too many: $\prod_{m\in\mathbb{F}_p}g(x+m)$ would be a factor of $f$ but it has degree $p\deg g>p$. So $g(x)=g(x+m)$ for some $m\in\mathbb{F}_p^\times$, but that would mean $g(x+n)$ for all $n\in\mathbb{F}_p$ are all equal, so $g$ has degree $\geq p$ (and divisible by $p$) by counting roots.
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