Example 3

Tag 06PK, permanent. Cite it as [Bourbaki, Tag 06PK].

Algebra, chapter V, § 11. Abelian extensions

Conversely, if a \in K - \mathfrak{p}(K), then the polynomial X^p - X - a is irreducible and the subfield L of \Omega generated by its roots is a cyclic extension of K of degree p. If a \in \mathfrak{p}(K), then X^p - X - a = \prod_{a \in \mathfrak{p}^{-1}(a)} (X - \alpha).

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

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