Theorem 3 (Jacobson)

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

Algebra, chapter V, § 13. p-radical extensions of height _ 1

— The mappings $K \mapsto g(K)$ and $l \mapsto I(h)$ are bijections of $\mathscr{C}$ onto $\mathscr{L}$ and of $\mathscr{L}$ onto $\mathscr{C}$ respectively, which are mutually inverse. If $K \in \mathscr{C}$ and $h \in \mathscr{L}$ correspond to each other, then $[E : K] = p^{[h : E]}$ .

The proof requires several preliminary lemmas.

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

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