Normal Subgroups and Galois Extensions

Abstract Algebra · Axiom Academy

LESSON Normal Subgroups and Galois Extensions The Fundamental Theorem's Crown Jewel: Understanding when intermediate extensions are Galois and how to construct extensions with prescribed Galois groups. For each subgroup H ≤ G, there's a fixed field K H = α ∈ K : σ(α) = α for all σ ∈ H This creates a correspondence: subgroups of G ↔ intermediate fields The correspondence reverses inclusions: H₁ ⊆ H₂ ⟺ K H₂ ⊆ K H₁ This theorem tells us precisely which intermediate extensions are Galois: exactly those corresponding to normal subgroups. And when they are Galois, their Galois group is the quotient group! 3. The Quotient Group Isomorphism When H ⊴ G, the restriction map gives us the isomorphism. Here's how it works: Well-defined: Since K/F is Galois, σ(K H ) ⊆ K H Surjective: By the Fundamental Theorem Kernel: ker(φ) = σ ∈ G : σ| K H = id = Gal(K/K H ) = H 4. Classic Example: ℚ(√2, √3)/ℚ Let K = ℚ(√2, √3) and F = ℚ. Then G = Gal(K/F) ≅ ℤ₂ × ℤ₂, generated by: All subgroups of ℤ₂ × ℤ₂ are normal (abelian group!), so all intermediate extensions are Galois: 5. Applications: Building Extensions This theorem is powerful for constructing extensions with prescribed Galois groups : Find a larger Galois extension K/F with Galois group H containing G Find a normal subgroup N ⊴ H such that H/N ≅ G Then K N /F is Galois with Gal(K N /F) ≅ H/N ≅ G

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