Example 4

Suppose that $K$ is of characteristic $p \neq 0$ and let $K(T)$ be the field of rational fractions and $U = T^p - T$. Put $E = K[U]$ and $F = K[T]$, then the polynomial $f(X) = X^p - X - U$ of $E[X]$ has the roots $T, T+1, ..., T+p-1$ in $F$. Let $\sigma$ be the $K$-automorphism of $F$ such that $\sigma(T) = T+1$. We have $\sigma^i(T) = T+i$ and $\sigma(U) = U$. The group $G = \{ 1, \sigma, ..., \sigma^{p-1} \}$ is cyclic of order $p$, and its field of invariants contains $E$; since $[F : E] \leq p$, Dedekind's theorem (V, p. 27, Cor. 2) implies that $E$ is the field of invariants of $G$ and $[F : E] = p$. The polynomial $f$ is thus irreducible in $E[X]$; the extension $F$ of $E$ is Galois, its Galois group $G$ is cyclic of order $p$, and the group $\Gamma$ is the group of cyclic permutations of $T, T+1, ..., T+p-1$.

For a generalization of this example see V, p. 93, Example 2.