Field theory MOC

Fixed field of an automorphism group

Let 𝐹 :𝐾 be a field extension and 𝐺 ≤Aut⁡(𝐹 :𝐾) be a group of field extension automorphisms. The fixed field of 𝐺 is the intermediate field field

𝐹𝐺:={𝛼∈𝐹:(∀𝑔∈𝐺)[𝑔𝛼=𝛼]}

The induced correspondence from intermediate fields 𝐹 :𝐸 :𝐾 to subgroups of 𝐺 ≤Aut⁡(𝐹 :𝐾) is called the Galois correspondence, which is an example of a Galois connection in that it is order-reversing:

𝐹𝐺≤𝐹𝐻⟺𝐻≤𝐺.

Moreover, for any 𝐺 ≤Aut⁡(𝐹 :𝐾) and 𝐹 :𝐸 :𝐾,

𝐸≤𝐹Aut⁡(𝐹:𝐸),𝐺≤Aut⁡(𝐹:𝐹𝐺);

Further still, if 𝐺1,𝐺2 ≤Aut⁡(𝐹 :𝐾) and 𝐹 :𝐸1,𝐸2 :𝐾,

Aut⁡(𝐹:𝐸1𝐸2)=Aut⁡(𝐹:𝐸1)∩Aut⁡(𝐹:𝐸2),𝐹⟨𝐺1,𝐺2⟩=𝐹𝐺1∩𝐹𝐺2;

where 𝐸1𝐸2 and ⟨𝐺1,𝐺2⟩ are the generated fields and groups respectively.

Proof

The only of these which is not immediate are the last ones. Suppose 𝜎 ∈Aut⁡(𝐹 :𝐸1) ∩Aut⁡(𝐹 :𝐸2). Then 𝜎 must also fix anything which can be written as a product of elements in 𝐸1 ands 𝐸2, hence 𝜎 ∈Aut⁡(𝐹 :𝐸1𝐸2), and therefore Aut⁡(𝐹 :𝐸1) ∩Aut⁡(𝐹 :𝐸2) ≤Aut⁡(𝐹 :𝐸1𝐸2). Since 𝐸1,𝐸2 ≤𝐸1𝐸2, the other inclusion is immediate.

Similarly, suppose 𝛼 ∈𝐹𝐺1 ∩𝐹𝐺2. Then 𝛼 must be fixed by any product of elements from 𝐺1 and 𝐺2, hence 𝛼 ∈𝐹⟨𝐺1,𝐺2⟩. Since 𝐺1,𝐺2 ≤⟨𝐺1,𝐺2⟩, the other inclusion is immediate.

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