Remark

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

Algebra, V, p. 69, § 10. Galois extensions

With the notation of Cor. 6 suppose that conditions a) and b) hold. The restriction homomorphisms $\varphi_i : \Gamma \to \mathrm{Gal}(E_i/K)$ for $i = 1, 2$ induce topological group isomorphisms

\Psi_1 : \Gamma_2 \to \mathrm{Gal}(E_1/K) , \quad \Psi_2 : \Gamma_1 \to \mathrm{Gal}(E_2/K)

By a) we see that the mapping

\sigma \mapsto (\varphi_1(\sigma), \varphi_2(\sigma))

is a topological group isomorphism of

\mathrm{Gal}(N/K)

onto

\mathrm{Gal}(E_1/K) \times \mathrm{Gal}(E_2/K)

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

Cited by

Nothing.

Cites

Cor. 6