Galois Theory Summary

Abstract Algebra · Axiom Academy

Let's review how field automorphisms create a bridge between group theory and field theory, revealing the deep structure behind polynomial solvability. Field Automorphisms & Galois Groups Field Automorphism: A bijective field homomorphism from a field to itself. These are the "symmetries" of field structure Galois Group: Gal(K/F) consists of all automorphisms of K that fix every element of F. Forms a group under composition Key Constraint: Automorphisms must send roots to roots. An automorphism σ maps α to another root of its minimal polynomial over F Size Bound: |Gal(K/F)| ≤ [K:F] , with equality precisely when K/F is Galois Normal Extension: Every irreducible polynomial over F that has one root in K has all its roots in K. Equivalently, K is a splitting field Separable Extension: Minimal polynomials have no repeated roots. Automatic in characteristic 0, requires care in positive characteristic Galois Extension: Both normal and separable. Equivalently, |Gal(K/F)| = [K:F] . These are the "well-behaved" extensions for Galois theory Splitting Fields: The splitting field of any separable polynomial is Galois. This includes cyclotomic extensions ℚ(ωₙ) Solvable Group: A group G is solvable if it has a composition series with abelian quotients. Think of it as "built from abelian pieces" Examples: All abelian groups are solvable. Sₙ is solvable for n ≤ 4 but not for n ≥ 5. The alternating group A₅ is simple and non-abelian

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