The Galois Group

Abstract Algebra · Axiom Academy

Understanding Gal(K/F) as the group of automorphisms that preserve the base field An automorphism σ is a bijective field homomorphism from K to itself. The condition σ(a) = a for all a ∈ F means σ "fixes" the base field F. These automorphisms capture the internal symmetries of the extension. Gal(K/F) forms a group under composition of functions. If σ and τ are automorphisms fixing F, then so is their composition σ ∘ τ. Closure: Composition of two automorphisms is an automorphism Identity: The identity map id: K → K fixes everything Inverses: Every automorphism is bijective, so invertible Associativity: Function composition is always associative A fundamental result: the size of the Galois group is bounded by the degree of the field extension. This makes intuitive sense: an automorphism is determined by where it sends a basis of K over F. Since there are [K:F] basis elements, and each must map to a conjugate, there are limited possibilities. 4. Quadratic Extension Example Consider the classic example: ℚ(√2) over ℚ. This is a degree 2 extension with [ℚ(√2):ℚ] = 2. Here we have |Gal(ℚ(√2)/ℚ)| = 2 = [ℚ(√2):ℚ], so equality holds! This extension is special - it's what we call a Galois extension . The magic question: when is |Gal(K/F)| = [K:F]? This happens precisely when K/F is a Galois extension . K is the splitting field of a separable polynomial over F Equivalently: K/F is normal and separable Equivalently: |Gal(K/F)| = [K:F]

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