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## Homework Statement

Let K be a Galois extension of F. Two intermediate fields E and L of field F are said to be conjugate if there exists

[tex]\sigma\in\text Gal_F K [/tex] such that [tex]\sigma (E) = L[/tex].

Prove that E and L are conjugates of F if and only if [tex]\text Gal_E K[/tex] and [tex]\text Gal_L K[/tex] are conjugate subgroups of [tex]\text Gal_F K[/tex].

## The Attempt at a Solution

From left to right, I have it already. I can't figure out how to get anywhere going from the right to left part of the proof.

I want to show that [tex]\sigma (E)=L [/tex] for some [tex]\sigma\in Gal_F K[/tex].

Let [tex]\alpha\in Gal_L K , \beta\in Gal_E K[/tex].

Since [tex]Gal_L K, Gal_E K [/tex]are conjugates, then [tex] Gal_E K=\{\sigma\alpha\sigma^{-1} | \alpha\in Gal_L K\}[/tex] or [tex] Gal_L K=\{\sigma^{-1}\beta\sigma | \beta\in Gal_E K\}[/tex].

[tex]\alpha=\sigma\beta\sigma^{-1}[/tex]

[tex]\beta=\sigma^{-1}\alpha\sigma[/tex]

I know that [tex]\beta[/tex] fixes E and that [tex]\alpha[/tex] fixes L.

[tex]\alpha (L)=\sigma\beta\sigma^{-1} (L)[/tex]

[tex]\beta (E)=\sigma^{-1}\alpha\sigma (E)[/tex]

[tex]L=\sigma\beta\sigma^{-1} (L)[/tex]

[tex]E=\sigma^{-1}\alpha\sigma (E)[/tex]

[tex]\sigma^{-1} (L)=\beta (\sigma^{-1} (L))[/tex]

[tex]\sigma (E)=\alpha (\sigma (E))[/tex]

[tex]\beta [/tex] fixes [tex]\sigma^{-1} (L)[/tex] and [tex]\alpha[/tex] fixes [tex]\sigma (E)[/tex]

[tex]\beta\in Gal_{\sigma^{-1} (L)} K[/tex] and [tex]\alpha\in Gal_{\sigma (E)} K[/tex]

From earlier: [tex]\beta\in Gal_E K[/tex] and [tex]\alpha\in Gal_L K[/tex]

So the only conclusion that I have is that the [tex]Gal_{\sigma (E)} K \cap Gal_L K [/tex] and [tex] Gal_{\sigma^{-1} (L)} K \cap Gal_E K[/tex] are nontrivial.

I feel like I should be using the Galois extension, i.e. Galois correspondence, to my advantage here, but I just don't see how it is applicable. It gives me that there exists a isomorphism from the intermediate fields to their respective Galois subgroup, i.e. [tex]\tau (E)=Gal_E K[/tex]. And since isomorphisms are order preserving, I get [tex]\left| {Gal_E K} \right|=\left| {E} \right|[/tex] and [tex]\left| {Gal_L K} \right|=\left| {L} \right|[/tex].

Any direction on the problem is appreciated.

Thanks in advance.

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