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I am trying to prove the following statement: Let $L/K$ be a Galois extension with $Gal(L/K)\cong G$. Consider the subgroups $H,N \subset G$. Then $$L_{H\cap N}=L_H\cup L_N$$ where $L_H$ denotes the fixed field.

Here's what I have so far: the Fundamental Theorem of Galois Theory says that since $H,N \supset N\cap H$, $L_{N\cap H}\supset L_N,L_H\Rightarrow L_{N\cap H}\supset L_N\cup L_H$.

What I'm struggling to show is that $L_{N\cap H}\subset L_N\cup L_H$. If an element of $N\cap H$ fixes some element of $L$, surely there's no guarantee this would also be fixed by either all of $N$, or all of $H$?

user404920
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    I think the end your first paragraph is supposed to read "where $L_H$ denotes..." since you use the subscript notation. – Arturo Magidin Feb 17 '21 at 21:52
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    Have you already established the bijective, inclusion-reversing correspondence between subgroups and intermediate fields? Also, I'm guessing $L_N\cup L_H$ is supposed to be the intermediate field generated by the union, rather than the literal union? – Arturo Magidin Feb 17 '21 at 21:53
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    As Arturo suggest, the notation $L_H \cup L_N$ is weird: that should be the composite field, usually written as $L_HL_N$. A union of subfields is not a subfield (unless one is contained in the other). If you know about the Galois correspondence think about the composite as the "smallest field containing $L_H$ and $L_N$" and then think about what happens to "smallest" and "containing" on the subgroup side when you apply the Galois correspondence. – KCd Feb 17 '21 at 21:58
  • Oh, I see! I misunderstood the compositum. So, would the correct argument be: $L_NL_H$ is the smallest subfield containing $L_H$ and $L_N$, so it corresponds to the largest subgroup of $G$ contained in $H$ and $N$. which is $H\cap N$? – user404920 Feb 17 '21 at 22:10
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    @user404920: Exactly. Similarly, $L_{\langle H,N\rangle} = L_H\cap L_N$. – Arturo Magidin Feb 17 '21 at 22:14

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