Separable Extensions

Abstract Algebra · Axiom Academy

Understanding when field extensions have "nice" behavior: no repeated roots, and the beautiful Primitive Element Theorem. 1. What Makes an Extension Separable? The key intuition: separability means "all roots are distinct." When a polynomial has repeated roots, something pathological is happening—and that only occurs in characteristic p. 2. The Gift of Characteristic 0 In fields of characteristic 0 (like ℚ, ℝ, ℂ), separability comes for free. Every algebraic extension is automatically separable—there's no way to create repeated roots! 3. The Trouble with Characteristic p In finite characteristic, things get interesting. The derivative can vanish, allowing inseparable polynomials to exist. The Frobenius map φ: x ↦ x^p is a homomorphism in characteristic p. A polynomial like x^p - a factors as (x - α)^p in the splitting field, where α^p = a. 4. The Primitive Element Theorem One of the most elegant results in field theory: finite separable extensions are simple , meaning they're generated by a single element. The Construction: For a finite extension, take θ = α₁ + cα₂ for generic c ∈ F. The separability guarantees that for almost all choices of c, we have F(θ) = F(α₁, α₂). Extending this gives the full result.

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