Building the Splitting Field of x³ - 2

Abstract Algebra · Axiom Academy

EXAMPLE Building the Splitting Field of x³ - 2 Step-by-step construction of Q(∛2, ω) through field adjunctions and degree computations. Excellent work! You've successfully constructed the splitting field of x³ - 2. Here's what we learned: Splitting fields require all roots: We needed both ∛2 and ω to capture all three roots: ∛2, ω∛2, and ω²∛2. Sequential adjunctions: We built Q(∛2, ω) by first adjoining ∛2 (degree 3), then ω (degree 2 over Q(∛2)). Tower law is essential: [Q(∛2, ω) : Q] = [Q(∛2, ω) : Q(∛2)] · [Q(∛2) : Q] = 2 · 3 = 6. Basis structure: The 6-dimensional basis combines elements from both extensions: 1, ∛2, ∛4, ω, ω∛2, ω∛4 . Minimal polynomials matter: x³ - 2 is minimal over Q, and x² + x + 1 is minimal for ω over Q(∛2). This construction demonstrates fundamental field theory concepts: how to systematically build extensions, compute degrees, and find bases. These techniques apply to constructing splitting fields of any polynomial!

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