f=x^p-x-a in a field of characteritic p

\mathbf{Problem:} Let F be of characteritic p and let a\in F. Show that f(x)=x^p-x-a has no multiple roots and that f(x) is irreducible in F[x] if and only if a\neq c^p-c for any c\in F.
\mathbf{Proof:} f'(x)=-1, so (f(x),f'(x))=1 which means f(x) is separable.
For the second part. If f is irreducible then immediately it has no roots in F. Conversely, suppose c is a root of f in some splitting field, c is not in F. Then by Freshman’s dream, one can verify c+1,c+2,\cdots, c+p-1 are also roots of f.

\displaystyle f(x)=(x-c)(x-c-1)\cdots(x-p+1)

The irreducible polynomial g(x) of c is a factor of f(x). g(x) must contain all the roots of f(x), otherwise the sum of these roots will results in c\in F. So f(x)=g(x) is irreducible.
\mathbf{Remark:} Jacoboson, Algebra I, p234.

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