Example 2

Let L be a cyclic extension of K of degree p; then the group (\mathfrak{p}(L) \cap K)/\mathfrak{p}(K) is cyclic of order p. If a \in K is such that the residue class of a mod \mathfrak{p}(K) is a generator of this group, then L is K-isomorphic to K[X]/(X^p - X - a) and the Galois group Gal(L/K) consists of the p automorphisms mapping X to X + i, i \in \mathbf{F}_p.