Fundamental Theorem of Galois Theory

Abstract Algebra · Axiom Academy

LESSON Fundamental Theorem of Galois Theory The beautiful correspondence between intermediate fields and subgroups that reveals the deep structure of field extensions. 1. The Setup: Galois Extensions Let K/F be a finite Galois extension with Galois group G = Gal(K/F) . This means: The correspondence reverses inclusions : E₁ ⊆ E₂ ⟺ Gal(K/E₁) ⊇ Gal(K/E₂) Degree formula : [K:E] = |Gal(K/E)| = |H| Index formula : [E:F] = [G : H] E/F is Galois ⟺ H is normal in G The correspondence maps each intermediate field E to the subgroup H = Gal(K/E) of automorphisms that fix E. Conversely, each subgroup H gives the fixed field E = K H = x ∈ K : σ(x) = x for all σ ∈ H . For an intermediate field E with corresponding subgroup H = Gal(K/E), the tower of fields F ⊆ E ⊆ K gives us crucial degree relationships: [K:E] = |H| — The degree of K over E equals the order of the subgroup H [E:F] = [G:H] — The degree of E over F equals the index of H in G [K:F] = [K:E]·[E:F] translates to |G| = |H|·[G:H] 5. Concrete Example: ℚ(√2, √3)/ℚ Consider K = ℚ(√2, √3) over F = ℚ. This is Galois with degree [K:ℚ] = 4 and Galois group G ≅ ℤ₂ × ℤ₂. H₁ = id, τ fixes √2, so K H₁ = ℚ(√2) H₂ = id, σ fixes √3, so K H₂ = ℚ(√3) H₃ = id, στ fixes √6, so K H₃ = ℚ(√6)

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