Solvability by Radicals — Mathematics Knowledge System

Overview

A polynomial equation is solvable by radicals if its roots can be expressed starting from the coefficients using only the field operations (+, −, ×, ÷) together with taking nth roots, exactly the way the quadratic formula solves ax²+bx+c=0 using a single square root. Galois's landmark theorem translates this analytic-looking question into a purely group-theoretic one: a polynomial is solvable by radicals over a field of characteristic 0 if and only if its Galois group is a solvable group — one built up from abelian pieces via a subnormal series. Since the symmetric group S₅ (and every Sₙ for n ≥ 5) is not solvable, this immediately proves the Abel–Ruffini theorem: there is no general algebraic formula, analogous to the quadratic formula, for the roots of a general polynomial of degree 5 or higher.

Intuition

Every step in a 'radical formula' — take a square root here, a cube root there — corresponds to adjoining a root of unity or a root of some element, which are exactly the kinds of extensions with abelian (in fact cyclic) Galois groups. Chaining several such steps builds the field up in a tower where each stage's Galois group is abelian relative to the one below it — precisely the definition of a solvable group. So 'the roots can be written with nested radicals' and 'the Galois group can be broken into abelian layers' turn out to be the same statement viewed from two different angles: one about formulas, one about group structure. This is why the quadratic, cubic, and quartic formulas exist (their Galois groups, subgroups of S₂, S₃, S₄, are always solvable) while no quintic formula exists — S₅ contains the simple, nonabelian group A₅ as its only proper nontrivial normal subgroup, so the subnormal series {e} ◁ A₅ ◁ S₅ can never be refined into abelian factors.

Formal Definition

Definition

Let K be a field of characteristic 0, and let f(x) ∈ K[x] have splitting field L.

K = F_0 \subset F_1 \subset \cdots \subset F_m, \qquad F_{i+1} = F_i(\sqrt[n_i]{a_i}) \text{ for some } a_i \in F_i

Radical tower: each step adjoins an n_i-th root

f \text{ is solvable by radicals over } K \iff \text{all roots of } f \text{ lie in some radical tower } F_m \text{ over } K

Definition of solvable by radicals

G \text{ is a solvable group} \iff \exists\, \{e\} = G_0 \trianglelefteq G_1 \trianglelefteq \cdots \trianglelefteq G_k = G, \quad G_{i+1}/G_i \text{ abelian}

Solvable group: subnormal series with abelian factors

f \text{ solvable by radicals} \iff \operatorname{Gal}(L/K) \text{ is a solvable group}

Galois's Theorem

Properties

Sₙ is solvable for n ≤ 4

S_2, S_3, S_4 \text{ are all solvable groups (subnormal series with abelian quotients exist), which is why the quadratic, cubic, and quartic formulas exist.}

Sₙ is not solvable for n ≥ 5

\text{For } n \geq 5,\ A_n \text{ is simple and nonabelian, so } S_n \text{ has no subnormal series with all-abelian factors} \implies S_n \text{ is not solvable.}

Abel–Ruffini theorem

\text{There is no general formula using } +,-,\times,\div,\sqrt[n]{\cdot}\ \text{ expressing the roots of a general degree-} n \text{ polynomial for } n \geq 5.

Subgroups and quotients of solvable groups are solvable

\text{If } G \text{ is solvable, every subgroup and every quotient of } G \text{ is also solvable} \text{ — used to rule out solvability by exhibiting } A_5 \text{ or } S_5 \text{ inside a Galois group.}

Worked Examples

Take the alternating subgroup A₃ ≤ S₃, which has order 3 (index 2, hence normal in S₃).
$A_3 \trianglelefteq S_3, \quad |A_3| = 3,\ [S_3:A_3]=2$
A₃ ≅ ℤ₃ is cyclic (abelian), and the quotient S₃/A₃ has order 2, so it is also abelian (ℤ₂).
$A_3 \cong \mathbb{Z}_3 \ (\text{abelian}), \quad S_3/A_3 \cong \mathbb{Z}_2 \ (\text{abelian})$
The series {e} ◁ A₃ ◁ S₃ has both factors A₃/{e} ≅ ℤ₃ and S₃/A₃ ≅ ℤ₂ abelian, satisfying the definition of solvable.
$\{e\} \trianglelefteq A_3 \trianglelefteq S_3$

Answer: S₃ is solvable via {e} ◁ A₃ ◁ S₃ with abelian factors ℤ₃ and ℤ₂, confirming that x³−2 = 0 is solvable by radicals (matching the classical cubic formula).

Practice Problems

Difficulty 7/10

Is S₄ (the Galois group possible for a general quartic) solvable? Give a subnormal series.

Difficulty 8/10

A specific quintic f(x) = x⁵ − 6x + 3 is known to have Galois group S₅ over ℚ (it has exactly 2 non-real roots, forcing a transposition together with a 5-cycle, which generate S₅). Is it solvable by radicals?

Difficulty 9/10

Explain why the existence of the quadratic, cubic, and quartic formulas, but not a quintic formula, is not a coincidence but a direct consequence of group theory.

Common Mistakes

Common Mistake

Thinking 'not solvable by radicals' means a polynomial has no roots.

Every polynomial still has all its roots in ℂ (Fundamental Theorem of Algebra); insolvability by radicals only means those roots cannot be written as a finite nested-radical expression in the coefficients.

Common Mistake

Assuming every specific quintic is unsolvable by radicals.

Only the GENERAL quintic (with Galois group S₅) is unsolvable; specific quintics with smaller (solvable) Galois groups, like x⁵−1 (cyclotomic, abelian group), are perfectly solvable by radicals.

Common Mistake

Confusing 'solvable group' with 'abelian group'.

Solvable is a strictly weaker condition than abelian — S₃ and S₄ are solvable but non-abelian, which is exactly why the cubic and quartic formulas exist despite involving genuinely non-commuting symmetries.

Quiz

Summary

A polynomial is solvable by radicals when its roots can be built from the coefficients using +,-,×,÷ and nth roots, corresponding to a tower of radical field extensions.
Galois's theorem: f is solvable by radicals over a characteristic-0 field iff Gal(splitting field/K) is a solvable group (subnormal series with abelian factors).
S₂, S₃, S₄ are all solvable, explaining the classical quadratic, cubic, and quartic formulas via explicit series like {e}◁A₃◁S₃ or {e}◁V₄◁A₄◁S₄.
Sₙ is not solvable for n ≥ 5 because Aₙ is simple and nonabelian, which is the precise reason the Abel–Ruffini theorem rules out a general quintic formula.
Individual quintics can still be solvable by radicals if their specific Galois group happens to be solvable (e.g. cyclic for cyclotomic-type polynomials) — only the GENERIC quintic (Galois group S₅) is provably unsolvable.

References

WebsiteWikipedia — Abel–Ruffini theorem
WebsiteWikipedia — Solvable group