The interesting thing is that the stabilizer of that automorphism is the base field, R. And this is a general phenomenon.
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Start with a field F, create an extension field E by adjoining some elements r_i. Now look at the automorphism group G(E/F).
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(If everything is "nice") the subgroups of G(E/F) correspond to intermediate fields between E and F, depending on which r_i they stabilize.
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Replying to @St_Rev @drossbucket
Starting from G(C/R) gives an intuition I didn’t remember and may or may not have gotten at the time…
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That doesn’t give an intermediate field; https://en.wikipedia.org/wiki/Fundamental_theorem_of_Galois_theory … uses Q(√2,√3) which is quite intuitive
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Replying to @Meaningness @drossbucket
Yeah traditionally introduced with a field s.t. G(E/F) = S_3, but C over R is a lot more intuitive to a lot more people w/less time.
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The full theorem also says "notice that larger subgroup = more automorphisms = smaller stabilizer subfield" so galois correspondence...
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...is actually an anti-isomorphism between the subgroup poset and the poset of intermediate fields.
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Replying to @St_Rev @drossbucket
I was going to say to
@drossbucket last night that “I think there was a picture and if you get the picture you get the idea,” and >1 reply 0 retweets 0 likes -
I think the mental picture of the two lattices being upside-down from each other is what I was failing to remember
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Also I have an (inaccurate but helpful) picture of the extensions lattice rising in a third dimension from a 2D base field
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I guess C is the prototypical field for me because I always think of them as being flat (which is obviously misleading at times)
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