Verifying Q(ω)/Q is Galois

Abstract Algebra · Axiom Academy

EXAMPLE Verifying Q(ω)/Q is Galois Step-by-step verification using cyclotomic extensions with n = 5 Excellent work! You've verified that Q(ω)/Q is a Galois extension. Here's what we established: Splitting Field: Q(ω) is the splitting field of x n - 1 over Q, containing all nth roots of unity Degree Formula: [Q(ω):Q] = φ(n), where φ is Euler's totient function counting integers coprime to n Galois Group Structure: Gal(Q(ω)/Q) ≅ (ℤ/nℤ)*, which is abelian and has order φ(n) Automorphisms: Each automorphism σ k is determined by σ k (ω) = ω k where gcd(k,n) = 1 General Result: This construction works for any n ≥ 1, giving a fundamental family of Galois extensions Cyclotomic extensions like Q(ω)/Q are cornerstones of Galois theory, with applications in algebraic number theory, cryptography, and the constructibility of regular polygons. The structure of (ℤ/nℤ)* determines the entire Galois group!

This is the written version of the interactive lesson above. See the full Abstract Algebra course.