Lucy Keer

Rev is so clear-thinking that maybe can give a 140 character explanation—or summarize in a few tweets… ?

Very wise mathematician once said, roughly: "In the beginning there were i and -i. But we don't know which is which. That's Galois theory."

Unpacking: We create C from R by adjoining the roots of x^2 + 1 = 0. This has two roots. Pick one, call it i, then the other is -i ...

...but we could have called either one i. That is, complex conjugation (sending i to -i) defines an automorphism C -> C.

The interesting thing is that the stabilizer of that automorphism is the base field, R. And this is a general phenomenon.

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).

(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.

Starting from G(C/R) gives an intuition I didn’t remember and may or may not have gotten at the time…