I'm reading through the proof in Dummit and Foote p. 593 that $$\operatorname{Gal}(K_1K_2/F) \cong H := \{(\sigma, \tau) \in \operatorname{Gal}(K_1/F) \times \operatorname{Gal}(K_2/F) \mid \sigma|_{K_1\cap K_2} = \tau|_{K_1\cap K_2}\}.$$

The idea is that the map \begin{align*} \operatorname{Gal}(K_1K_2/F) &\to H \\ \sigma &\mapsto (\sigma|_{K_1}, \sigma|_{K_2}) \end{align*} is an isomorphism. I understand that this map is injective, but in their argument for surjectivity, they say that "for every $\sigma\in \operatorname{Gal}(K_1/F)$, there are $|\operatorname{Gal}(K_2/K_1\cap K_2)|$ elements $\tau\in \operatorname{Gal}(K_2/F)$ whose restrictions to $K_1\cap K_2$ are $\sigma|_{K_1\cap K_2}$." This is not explained, and I'm having trouble understanding this computation.

Any help would be greatly appreciated. Thank you!

  • $\begingroup$ You can probably make a parallelism to the second isomorphism theorem. $\endgroup$ – Pedro Tamaroff Mar 15 '14 at 0:10

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