$x^p -x-c$ is irreducible over a field of characteristic $p$ if it has no root in the field

Solution 1:

Let $L$ be the splitting field and $\alpha \in L$ a root. Consider the map $Gal(L/F) \to \mathbb Z/p\mathbb Z$ given by $\sigma \mapsto \sigma(\alpha)-\alpha$ (You should verify that this is well defined). Since $\mathbb Z/p\mathbb Z$ has no non-trivial subgroups, the map is either trivial or surjective. If the map is surjective, we deduce that the galois group acts transitively on the roots, hence the polynomial is irreducible. If the map is trivial, we deduce $\alpha \in F$, which is a contradiction to the assumption.

Without Galois theory (but somehow the same proof): Consider the isomorphism $\varphi: F[x]\to F[x], x \mapsto x+1$. The polynomial $f = x^p-x-c$ is fixed by $\varphi$, hence the irreducible factors are permuted by $\varphi$. Let the irreducible factors be $M := \{f_1, \dotsc, f_n \}$. Since there is no root in $F$, we have $n < p$.

$\langle \varphi \rangle \cong \mathbb Z/p\mathbb Z$ acts on $M$, hence we get a homomorphism $\langle \varphi \rangle \cong \mathbb Z/p\mathbb Z \to S_n$. From $n<p$ we deduce that this homomorphism is trivial, hence the action is trivial, so all irreducible factors are fixed by $\varphi$. You should finish the proof by showing that $\varphi$ does not fix any polynomials of degree $1, 2 ,\dotsc, p-1$. Hence the only irreducible factor can be $f$ itself.

Solution 2:

It's equivalent to show $x^p-x-c$ is reducible in $F[x]$ $\implies$ $x^p-x-c$ has a root in $F$.

Suppose first that $u$ is the root of $x^p-x-c$ in some extension field $K \supset F$, then $u, u+1, \cdots, u+p-1$ are all its roots.

If $x^p-x-c$ is reducible in $F[x]$, then for $x-u, x-(u+1), \cdots, x-(u+p-1)$, there're $n \ge 1$ of them with their multiplication in $F[x]$.

The coefficient term of $x^{n-1}$ belongs to $F$, so $nu \in F$ and $u \in F$.