The Galois Correspondence

Example 1.1. Two R-automorphisms of C are the identity z 7→ z and complex conjugation z 7→ z. We will show they are the only ones. If σ : C→ C is an R-automorphism, then for any real a and b we have σ(a+ bi) = σ(a) +σ(b)σ(i) = a+ bσ(i), so σ is determined by σ(i) and i = −1 =⇒ σ(i) = σ(−1) =⇒ σ(i) = −1 =⇒ σ(i) = ±i. If σ(i) = i, then σ(z) = z for all z ∈ C and if σ(i) = −i, then σ(z) = z for all z ∈ C. From any intermediate field K ⊂ F ⊂ L we get a subgroup