Why Quintic Equations Have No General Formula | Galois Theory

In Galois theory, a polynomial can be solved by radicals if and only if its Galois group is a solvable group. A solvable group is one that can be split into abelian groups in some sense. The symmetric groups S₁ through S₄ are solvable, which explains why polynomials of degree four or lower can be solved by radicals. However, S₅ is not solvable, meaning there exist degree five polynomials whose Galois group is S₅ that cannot be solved by radicals.

This extensive section develops the connection between radical extensions and solvable polynomials. It begins by proving that if a field K contains all nth roots of unity and a is in K, then K(ⁿ√a) is a Galois extension with cyclic Galois group generated by multiplication by ζₙ. This establishes that radical extensions correspond precisely to cyclic Galois extensions. The section then introduces the solvability criterion: a polynomial is solvable by radicals if and only if its Galois group is solvable—having a chain of normal subgroups with cyclic quotients. This provides a complete characterization of solvable polynomials in terms of their Galois group structure, setting up the proof that quintic polynomials are generally unsolvable.

The existence of formulas for polynomial solutions follows a pattern: linear (degree 1) and quadratic (degree 2) equations have simple formulas, cubic (degree 3) has a more complex formula, quartic (degree 4) also has a formula though unwieldy, but no such formula exists for quintic (degree 5) and higher polynomials. This is related to the solvability of Galois groups associated with polynomials - a formula exists if and only if the Galois group is solvable. This research was conducted by Evariste Galois in the field of Galois theory.

Sn is solvable iff n ≤ 4. For n = 1,2,3: Sn' = An, An' = {e}. For n = 4: S4' = A4, A4' = V4 (abelian), V4' = {e}. For n > 4: S_n' = A_n, and A_n' = A_n, so we never reach the trivial subgroup. Galois theory connects this to polynomial solvability: a polynomial is solvable iff its Galois group is solvable. Since S_n is not solvable for n ≥ 5, there is no general solution formula using radicals for quintic and higher-degree polynomials.

Galois theory provides definitive answers to classical geometric construction problems. Angle trisection fails because the corresponding Galois group is cyclic of order three, which is not a power of two (the requirement for ruler-and-compass constructions). In contrast, Gauss proved at age 19 that the regular 17-gon is constructible because the automorphism group of the cyclic group of order 17 has order 16, a power of two. These examples demonstrate how field extensions and their Galois groups determine geometric possibility.

Évariste Galois developed group theory and Galois theory, which resolved the classical problems of antiquity. He proved that quintic equations cannot be solved by radicals (no formula exists like the quadratic formula). This explained why trisection of angles and doubling cubes are impossible with compass and straightedge. In 1882, Lindemann proved pi is transcendental, demonstrating that squaring the circle is also impossible.

Beyond doubling the cube, Greeks obsessed over squaring the circle (constructing a square with the same area as a given circle) and trisecting an angle (splitting an arbitrary angle into three equal parts). Greeks likely assumed these problems were unsolvable with ruler and compass, so elaborate solutions proposed alternate construction methods. It took until 1837 for Pierre Wantzel to prove doubling the cube and trisection are impossible with ruler and compass using Galois theory. Ferdinand von Lindemann finally proved in 1882 that squaring the circle is impossible because pi is transcendental (never a solution to a polynomial equation).

This section presents the complete theory of constructibility: a number β is constructible if and only if it belongs to a field extension K₀(α₁, ..., αₙ) where K₀ contains the given values, and each extension is obtained by adjoining a square root. This means constructible numbers are those reachable by repeatedly taking square roots. The video explains that this theory proves the impossibility of the three classical Greek construction problems: squaring the circle (requires √π, which involves π, not constructible), doubling the cube (requires 2^(1/3), which is not a square root extension), and angle trisection (requires solving cubic equations, not square root extensions). This demonstrates the power of Galois theory in solving classical geometric problems.

Galois Theory is a branch of algebra that studies the solvability of polynomial equations by examining the symmetries between their roots, establishing a profound connection between field extensions and group theory; it answers fundamental questions about whether polynomials of degree 5 or higher can be solved with general formulas and provides elegant proofs for classical geometric construction problems like the impossibility of squaring the circle or trisecting an arbitrary angle using only a compass and straightedge.

The Inverse Galois Problem asks whether every finite group can be realized as a Galois group over the rational numbers Q. This talk presents partial solutions: every finite abelian group is realizable via cyclotomic fields, the symmetric group S_n is realizable using the polynomial x^n - x - 1, and GL(2, F_3) is realizable through elliptic curve torsion points. Key tools include the Kronecker-Weber theorem, decomposition/inertia subgroups in algebraic number theory, and the Mordell-Weil theorem for elliptic curves. While many groups like alternating groups and solvable groups are known to be realizable, the problem remains open for certain groups such as the Mathieu group M23.

The Inverse Galois Problem asks whether every finite group can be realized as a Galois group over the rational numbers Q or over the rational function field Q(t). While this remains an open conjecture for general finite groups, significant progress has been made for specific classes of groups, including symmetric groups (proven by Hilbert using his Irreducibility Theorem) and all soluble groups (proven by Shafarevich). The problem connects deeply with algebraic geometry, number theory, and group cohomology, and computational tools like Magma help researchers construct and study these Galois extensions explicitly.

The inverse Galois problem asks whether every finite group can be realized as the Galois group of a Galois extension of the rational numbers, and while many groups (symmetric groups, alternating groups, solvable groups, and most sporadic simple groups) are known to be realizable, the general question remains open; geometric approaches using torsors and Hilbert's irreducibility theorem provide powerful methods for constructing such extensions, though limitations exist—for instance, Noether's problem has a negative answer for Z/8Z, demonstrating that not all faithful group actions on projective space yield rational quotients.

The Inverse Galois Problem asks whether every finite group can occur as a Galois group over the field of rational numbers Q; it has been proven that all finite abelian groups can be realized as Galois groups over Q by constructing appropriate polynomials whose splitting fields have the desired Galois group structure, while Shafarevich proved that every solvable group occurs as a global Galois group over Q, though these arguments are not always constructive.

An infinite Galois extension is the union of finite Galois extensions. The Galois group is the inverse limit of finite Galois groups. There is a bijection between intermediate fields and closed subgroups of the absolute Galois group. Normal subgroups correspond to normal extensions. Open subgroups correspond to finite extensions. The inverse Galois problem asks which finite groups can be realized as Galois groups over Q. All solvable groups can be realized, but it is unknown whether all finite groups can be realized. Over C(t), every finite group can be realized. This is a major open problem in algebraic number theory.

Abstract Galois theory studies profinite groups (compact, Hausdorff, totally disconnected topological groups that are projective limits of finite groups) as Galois groups without reference to specific field extensions, enabling a unified approach to class field theory through reciprocity maps that establish isomorphisms between Galois groups and idele class groups.

This section establishes core concepts in algebraic number theory. The field of algebraic numbers consists of complex numbers satisfying polynomial equations with rational coefficients. The absolute Galois group Gal(Q̄/Q) encodes all automorphisms of this field, forming a large profinite group. Galois representations map this group to GL_n(K), capturing arithmetic information. Quadratic reciprocity, solved by Gauss, determines when quadratic polynomials have solutions modulo primes. Class field theory generalizes this to all polynomials, describing how Galois groups act on roots. For Q and CM fields, all abelian extensions are obtained by adjoining roots of unity, though extending to general number fields remains an open problem.

Class field theory provides a systematic framework for understanding abelian extensions of number fields. An algebraic number is a root of a non-zero polynomial with rational coefficients. The Kronecker-Weber theorem classifies all abelian extensions of Q as composites of cyclotomic fields Q(ζ_n). For general number fields F, class field theory describes narrow ray class fields F_n that generalize cyclotomic fields, with nested structure and collectively containing all abelian extensions. The Galois groups are described via class groups and ray class groups. While class field theory describes Galois groups, explicit class field theory seeks to construct these fields explicitly. For quadratic imaginary fields, complex multiplication provides constructions using special values of modular functions evaluated at torsion points. Hilbert's 12th problem (1900) asked to extend Kronecker-Weber to arbitrary algebraic number fields, recognizing that special values of modular functions solved the imaginary quadratic case. Stark's conjectures generalize Dirichlet's class number formula, predicting special units in abelian extensions whose absolute values relate to L-functions. Tate unified Brumer's class group conjecture with Stark's unit conjecture into the Brumer-Stark conjecture, while Gross refined this with p-adic L-functions in the Gross-Stark conjecture. For totally real fields, Brumer-Stark units are special p-units in ray class fields with specific valuation properties, providing a path toward explicit class field theory for totally real fields.

In algebraic number theory, when unique factorization fails in certain number rings (such as Z[√-5] where 6 = 2×3 = (1+√-5)(1-√-5)), we can restore unique factorization by factoring ideals instead of numbers; the class number counts how many fundamentally different types of non-principal ideals exist in a ring, and Iwasawa Theory reveals that in infinite towers of cyclotomic fields, the p-part of the class number grows exponentially as p^λn + ν, with the μ-invariant being zero for all primes (proven by Ferrero and Washington in 1979).

This extensive section covers advanced topics in algebraic geometry and field theory. It examines cubic curves like the strophoid, demonstrating how singularities affect parametrizations and requiring different solution methods than conics. Fermat's innovative approach to solving cubic equations through tangent line iteration is presented, showing how geometric methods systematically find rational solutions. The section transitions to field extensions, defining algebraic extensions where every element satisfies a polynomial with coefficients in the base field. Key examples include i over ℝ and √2 over ℚ, contrasting with transcendental numbers like π and e. The lecture proves that finite extensions are always algebraic: if [K:k]=d, then 1, α, ..., α^d are linearly dependent, yielding a polynomial equation. Finally, it establishes that algebraic numbers form a field ℚ̄, the algebraic closure of ℚ, containing all roots of polynomials with rational coefficients.