Normal Extensions

Abstract Algebra · Axiom Academy

Field extensions where every irreducible polynomial with one root contains all its roots— the "all-or-nothing" property of algebraic extensions. In other words: if one root is in, all roots are in. There's no partial membership— it's all or nothing for conjugate roots of irreducible polynomials. Consider the polynomial x² - 2 over ℚ . It's irreducible and has roots ±√2. If we adjoin just √2 to get ℚ(√2), is this extension normal? Yes! Because √2 ∈ ℚ(√2) forces -√2 = -(√2) ∈ ℚ(√2) as well. The field operations automatically give us both roots. 3. Forcing Conjugates to Appear Here's where it gets interesting. Consider x³ - 2 over ℚ . Its roots are ∛2, ω∛2, and ω²∛2 (where ω = e^(2πi/3) is a primitive cube root of unity). ℚ(∛2) is NOT normal! It contains the real root ∛2, but not the complex roots ω∛2 and ω²∛2. The polynomial doesn't split completely. However, ℚ(∛2, ω) IS normal—once we have both ∛2 and ω, we get all three roots via field operations: ∛2, ω·∛2, and ω²·∛2. This equivalence is powerful. It means normal extensions are exactly those obtained by "completing" a polynomial—adding all the roots you need to factor it completely. Normal extensions are special because they're invariant under conjugation . Any field automorphism σ : K → K that fixes F must map roots to roots. Closure: Contains all conjugates of its elements Galois Connection: When also separable, becomes a Galois extension Compositum: Compositum of normal extensions is normal

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