Cyclotomic Extensions

Abstract Algebra · Axiom Academy

Exploring Q(ωₙ) and its Galois group (ℤ/nℤ)*, with applications to constructible regular polygons 1. Roots of Unity & Cyclotomic Extensions We typically take ωₙ = e^(2πi/n), which generates all nth roots of unity as powers: 1, ωₙ, ωₙ², ..., ωₙⁿ⁻¹. These roots lie equally spaced on the unit circle in the complex plane. The cyclotomic extension Q(ωₙ)/Q is a Galois extension—it's normal (splitting field of xⁿ - 1) and separable (characteristic 0). The Galois group has a beautiful structure. The group operation corresponds to multiplication mod n: σⱼ ∘ σₖ = σⱼₖ. This gives us the isomorphism with (ℤ/nℤ)*, the multiplicative group of units modulo n. 3. Why Cyclotomic Extensions Are Solvable A key property of (ℤ/nℤ)* is that it's always abelian: multiplication mod n is commutative. This immediately tells us something important about solvability. 4. Regular Polygon Constructibility A regular n-gon is constructible with compass and straightedge if and only if we can construct ωₙ = cos(2π/n) + i·sin(2π/n), which requires constructing cos(2π/n). 5. Examples: Constructible vs Non-Constructible Let's examine specific cases to see the criterion in action:

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