Kummer Extensions

Abstract Algebra · Axiom Academy

Understanding how adjoining nth roots creates beautifully structured field extensions when primitive roots of unity are present. A Kummer extension arises when we adjoin an nth root to a field that already contains a primitive nth root of unity ωₙ. Base field K contains ωₙ (primitive nth root of unity) gcd(n, char(K)) = 1 (characteristic doesn't divide n) Let α = ⁿ√a be one nth root of a. Because K contains ωₙ, all nth roots of a are given by α, ωₙα, ωₙ²α, ..., ωₙⁿ⁻¹α. The minimal polynomial of α over K divides xⁿ - a, and since we have all roots, the extension L/K is Galois (normal and separable). Any σ ∈ Gal(L/K) must send α to another nth root of a. Since ωₙ ∈ K, we have σ(ωₙ) = ωₙ, so σ is determined by where it sends α. We can adjoin multiple nth roots simultaneously. If L = K(ⁿ√a₁, ⁿ√a₂, ..., ⁿ√aₘ), the Galois group embeds into (ℤ/nℤ)ᵐ. The Kronecker-Weber theorem states that every finite abelian extension of ℚ is contained in a cyclotomic field ℚ(ωₙ) for some n. Start with ℚ(ωₙ), which contains all nth roots of unity Any abelian extension can be built by Kummer extensions Repeatedly adjoin nth roots to construct all abelian extensions The cyclotomic fields ℚ(ωₙ) serve as the "base" containing the necessary roots of unity

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