Radical Extensions

Abstract Algebra · Axiom Academy

Understanding field towers, solvability by radicals, and the profound connection to Galois theory Each extension in the tower is obtained by adjoining an nth root of an element from the previous field. Think of it as climbing a ladder where each rung adds a new radical. When we write Fᵢ₊₁ = Fᵢ(ⁿ√a), we're creating a new field that contains Fᵢ and also contains a solution to xⁿ - a = 0. This element satisfies (ⁿ√a)ⁿ = a. In other words, if we can build a radical tower that contains all the roots of the polynomial, then the polynomial is solvable by radicals. This is exactly what formulas like the quadratic formula do! Splitting field: K = ℚ(√2, -√2) = ℚ(√2) Since K ⊆ ℚ(√2), the polynomial is solvable by radicals ✓ 4. Connection to Galois Groups The profound theorem connecting radical extensions to Galois theory states: A group G is solvable if it has a subnormal series where each quotient is abelian: Radical extensions provide a beautiful bridge between three areas of mathematics: Algebraic: Field towers built by adjoining roots Polynomial: Solvability of equations by radical formulas Group-Theoretic: Solvable Galois groups Degrees 2, 3, 4: Always solvable by radicals (Galois groups are solvable) Degree 5+: Generic polynomials NOT solvable (S₅ and higher not solvable) Special cases in degree 5+: Some polynomials ARE solvable if their Galois group is solvable

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