Fact-checked by Grok 2 weeks ago

Abel–Ruffini theorem

The Abel–Ruffini theorem, also known as Abel's impossibility theorem, states that there is no general algebraic solution in radicals for polynomial equations of degree five or higher.^[1] This result resolved a longstanding question in algebra dating back to the Renaissance, when explicit formulas using radicals were discovered for quadratic, cubic, and quartic equations.^[1] Italian mathematician Paolo Ruffini first claimed the impossibility for quintics in 1799, providing an incomplete proof that covered specific cases but lacked generality.^[1] Norwegian mathematician Niels Henrik Abel established the full theorem in 1824 with a rigorous six-page proof, demonstrating that no such radical formula exists for the general fifth-degree polynomial.^[2] Independently, French mathematician Évariste Galois developed a broader framework in 1832—published posthumously in 1846—that not only confirmed Abel's result but also provided criteria for solvability by radicals for any polynomial degree, via the concept of solvable Galois groups.^[1] The theorem's proof relies on Galois theory, which associates a polynomial's roots to the structure of its Galois group; for the general quintic, this group is the symmetric group S_5, which is not solvable, implying no radical expression for the roots.^[1] While specific quintics (e.g., x^5 - 1 = 0) remain solvable by radicals, the theorem underscores the limitations of elementary operations for arbitrary coefficients, profoundly influencing modern algebra and group theory.^[1] Later proofs, such as Vladimir Arnold's 1963 topological approach using Riemann surfaces and monodromy, offer alternative perspectives without Galois theory, confirming the result via continuous functions and branching.^[3]

Overview and Statement

Theorem Formulation

The Abel–Ruffini theorem asserts that there is no general algebraic solution in radicals to polynomial equations of degree five or higher with arbitrary coefficients.^[4]^[5] Specifically, for the general polynomial equation p(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 = 0 where n \geq 5 and the coefficients a_i are arbitrary elements of the rational or real numbers, no formula exists to express the roots using only the coefficients and a finite sequence of the field operations of addition, subtraction, multiplication, division, and extraction of roots.^[4]^[5] A polynomial equation is solvable by radicals if all of its roots can be expressed in terms of its coefficients through a finite number of applications of the arithmetic operations (addition, subtraction, multiplication, and division) and the extraction of k-th roots for integers k \geq 2, potentially nested.^[4] This process corresponds to adjoining roots successively to the base field generated by the coefficients, forming a tower of radical extensions.^[4] In contrast, polynomials of degree one through four admit explicit solutions by radicals: the linear case is direct, the quadratic formula provides a solution via square roots, and cubic and quartic equations have formulas involving cube roots and fourth roots, respectively, though more complex.^[4]^[5] The theorem applies particularly to irreducible polynomials over the rationals or reals with generic coefficients, meaning those for which the Galois group acts transitively and without special symmetries that might allow radical solutions in particular cases.^[4] While specific polynomials of degree five or higher may be solvable by radicals—for instance, if they factor or have solvable Galois groups—the result establishes that no universal radical formula exists for the general case, marking a fundamental limitation in algebraic solution methods.^[4]^[5]

Historical Significance

The Abel–Ruffini theorem represented a profound shift in algebraic research, moving from the centuries-long pursuit of explicit formulas using radicals—exemplified by Cardano's solution for cubic equations in 1545 and Ferrari's for quartics around 1540—to the recognition of fundamental limitations through impossibility proofs. This change ended the "quest for formulas" that had dominated mathematics since ancient times, redirecting efforts toward understanding the structural properties of equations rather than constructing solutions.^[5] By resolving a problem that had persisted for nearly three centuries since the quartic solution, the theorem catalyzed the development of modern abstract algebra in the 19th century. Ruffini's partial attempt in 1799 and Abel's rigorous proof in 1824 laid foundational ideas about permutations and solvability, which directly influenced the emergence of group theory as a central mathematical discipline. These insights transformed algebra from a tool for computation into a field focused on symmetry and invariance, paving the way for Évariste Galois's more comprehensive framework later in the decade.^[5] The theorem's broader implications extended beyond algebra, profoundly affecting number theory by highlighting irreducibility in field extensions and geometry through connections to classical construction problems, such as the impossibility of certain ruler-and-compass tasks. In the long term, it spurred the adoption of numerical and computational methods for approximating roots of higher-degree polynomials, influencing 20th-century approaches to solving equations without closed forms. This structural perspective underscored by the theorem remains a cornerstone of contemporary mathematics.

Mathematical Prerequisites

Field Theory Basics

In field theory, a field F is a commutative ring with unity where every non-zero element has a multiplicative inverse, providing the foundational structure for algebraic extensions. The rational numbers \mathbb{Q} serve as a common base field for studying polynomial equations over the reals or complexes. An extension field K of F, denoted K/F, is a field containing F as a subfield, where elements of K not in F are adjoined to solve equations or factor polynomials. The degree of the extension [K:F] is the dimension of K as a vector space over F, which is finite if K is generated by finitely many algebraic elements over F.^[6] The splitting field of a polynomial f(x) \in F over F is the smallest extension K/F in which f(x) factors completely into linear factors. This field is generated by adjoining all roots of f(x) to F and is unique up to isomorphism over F. For example, the splitting field of x^n - 1 over \mathbb{Q} is the nth cyclotomic field \mathbb{Q}(\zeta_n), where \zeta_n is a primitive nth root of unity, and its degree [\mathbb{Q}(\zeta_n):\mathbb{Q}] = \phi(n), with \phi denoting Euler's totient function. Cyclotomic fields are particularly relevant in the context of adjoining radicals, as roots of unity facilitate the extraction of nth roots in solvability discussions.^[7] An algebraically closed field is an extension where every non-constant polynomial has at least one root, implying that all polynomials factor into linear factors. The algebraic closure \overline{F} of a field F is the smallest algebraically closed field containing F, consisting of all elements algebraic over F. For instance, the complex numbers \mathbb{C} form an algebraic closure of \mathbb{R}, while the field of algebraic numbers \overline{\mathbb{Q}} is the algebraic closure of \mathbb{Q}, a proper subfield of \mathbb{C}. Every polynomial over F splits completely in \overline{F}.^[8] For a tower of extensions F \subseteq E \subseteq K, the tower law states that the degree [K:F] = [K:E] \cdot [E:F], provided the degrees are finite; this multiplicative property allows decomposition of complex extensions into simpler steps.^[9] A field extension K/F is normal if every irreducible polynomial in F with at least one root in K splits completely into linear factors within K. This ensures that conjugates of roots are also contained in K, making it closed under the action of embeddings into an algebraic closure. An extension is separable if every element of K is algebraic over F with a separable minimal polynomial, meaning its roots are distinct in an algebraic closure; in characteristic zero fields like \mathbb{Q}, all algebraic extensions are separable. Normal and separable extensions together form the basis for advanced structural theorems in algebra.^[10]

Solvability by Radicals

In field theory, a radical extension of a field K is formed by adjoining elements whose powers lie in the previous field in a tower. Specifically, an extension L/K is radical if there exists a finite sequence of fields K = K_0 \subseteq K_1 \subseteq \cdots \subseteq K_m = L such that for each i = 1, \dots, m, there is an element u_i \in K_i and an integer n_i \geq 2 with u_i^{n_i} \in K_{i-1}.^[11] A simple case is adjoining an nth root, where L = K(\sqrt{a}) for some a \in K and n \geq 2, satisfying [\sqrt{a}]^n = a \in K.^[12] A chain of radical extensions, or radical tower, builds successively by adding such roots at each step, ensuring the full extension is radical. For a polynomial f(x) \in K, solvability by radicals means its splitting field \Sigma over K is contained in some radical extension M/K. This containment implies \Sigma \subseteq M, where M arises from a finite radical tower over K.^[13] Such a finite radical tower corresponds directly to solution formulas for the roots of f(x), expressible using elements of K combined through addition, subtraction, multiplication, division, and extraction of nth roots for various n. The operations in the tower mirror these arithmetic and radical extractions, yielding explicit radical expressions for the roots without requiring infinite processes or transcendental functions.^[14] A key result characterizes this solvability: a polynomial f(x) over a field K of characteristic zero is solvable by radicals if and only if its splitting field admits a radical extension tower over K. This equivalence bridges the algebraic structure of the splitting field to constructive solution methods via radicals.^[15] For illustration, consider a quadratic equation x^2 + bx + c = 0 over K, with roots given by the quadratic formula \frac{-b \pm \sqrt{b^2 - 4c}}{2}. The splitting field is K(\sqrt{b^2 - 4c})/K, a simple radical extension by adjoining the square root, demonstrating solvability by radicals for degree 2.^[16]

Proof Outline

Galois Theory Framework

The Galois group of a field extension K/F, denoted \mathrm{Gal}(K/F), is defined as the group of field automorphisms of K that fix F pointwise, provided that K/F is a Galois extension, meaning it is algebraic, normal, and separable.^[17]^[18] This group captures the symmetries of the extension, and for finite Galois extensions, its order equals the degree of the extension: |\mathrm{Gal}(K/F)| = [K:F].^[19]^[20] The fundamental theorem of Galois theory establishes a bijective correspondence between the subgroups of \mathrm{Gal}(K/F) and the intermediate fields between F and K. Specifically, for a subgroup H \leq \mathrm{Gal}(K/F), the fixed field K^H = \{ x \in K \mid \sigma(x) = x \ \forall \sigma \in H \} is an intermediate field, and conversely, for an intermediate field L with F \subseteq L \subseteq K, the subgroup \mathrm{Gal}(K/L) consists of those automorphisms fixing L. This correspondence is inclusion-reversing: larger subgroups correspond to smaller fixed fields, and normal subgroups correspond to intermediate fields that are themselves Galois over F.^[21]^[22]^[23] In the context of solving polynomial equations, consider an irreducible separable polynomial f(x) \in F of degree n with splitting field K over F. The Galois group \mathrm{Gal}(K/F) acts faithfully on the n roots of f(x) by permuting them, embedding \mathrm{Gal}(K/F) as a transitive subgroup of the symmetric group S_n. This action reflects the irreducibility of f(x), ensuring transitivity, and allows the group's structure to determine properties of the roots.^[24]^[25]^[26] A key application to the Abel–Ruffini theorem is the solvability criterion: the roots of f(x) can be expressed by radicals over F if and only if \mathrm{Gal}(K/F) is a solvable group, meaning it admits a composition series whose factor groups are abelian. This criterion translates the algebraic problem of radical solvability into a purely group-theoretic condition on the Galois group.^[27]^[28]^[15]

Structure of Symmetric Groups

The symmetric group S_n consists of all permutations of a set with n elements under composition, forming a group of order n!.^[29] This group is generated by the set of all transpositions, which are permutations that swap two elements and leave the others fixed.^[30] A fundamental subgroup of S_n is the alternating group A_n, which comprises all even permutations—those that can be expressed as a product of an even number of transpositions.^[31] For n \geq 3, A_n is a normal subgroup of S_n with index 2, meaning |S_n : A_n| = 2 and |A_n| = n!/2.^[32] A group is solvable if it possesses a composition series where each factor group is abelian. For small values of n, the symmetric groups exhibit such series. Specifically, S_2 is isomorphic to the cyclic group \mathbb{Z}/2\mathbb{Z}, which is abelian and thus solvable.^[33] For S_3, a composition series is given by \{e\} \trianglelefteq A_3 \trianglelefteq S_3, where A_3 \cong \mathbb{Z}/3\mathbb{Z} and the factors are A_3 / \{e\} \cong \mathbb{Z}/3\mathbb{Z} and S_3 / A_3 \cong \mathbb{Z}/2\mathbb{Z}, both abelian.^[34] Similarly, S_4 admits the composition series \{e\} \trianglelefteq V_4 \trianglelefteq A_4 \trianglelefteq S_4, where V_4 is the Klein four-group (isomorphic to \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}), yielding abelian factors V_4 / \{e\} \cong \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}, A_4 / V_4 \cong \mathbb{Z}/3\mathbb{Z}, and S_4 / A_4 \cong \mathbb{Z}/2\mathbb{Z}.^[35] In contrast, for n \geq 5, S_n is nonsolvable. The alternating group A_5 has order 60 and is simple, meaning its only normal subgroups are the trivial group and itself; moreover, it is nonabelian.^[36] Since A_5 is a normal subgroup of S_5 with quotient S_5 / A_5 \cong \mathbb{Z}/2\mathbb{Z}, the presence of the simple nonabelian composition factor A_5 implies that no composition series for S_5 can have all abelian factors.^[37] Equivalently, the derived series of S_5 fails to terminate at the trivial subgroup because the commutator subgroup of A_5 is A_5 itself, preventing further reduction to abelian groups.^[38] This nonsolvability extends to S_n for all n \geq 5, as each contains A_5 as a subgroup whose structure resists abelian decomposition.^[33]

Unsolvability for Degree Five and Higher

The unsolvability of the general polynomial equation of degree n \geq 5 by radicals is established by determining its Galois group over the field of rational functions in the coefficients, which is the symmetric group S_n. This group acts on the n roots of the polynomial f(x) = x^n + a_{n-1} x^{n-1} + \cdots + a_0, considered over \mathbb{Q}(a_0, \dots, a_{n-1}). The polynomial is irreducible over this base field, ensuring that the Galois group is transitive.^[21] A proof sketch proceeds by showing that S_n is generated within the Galois group using resolvent polynomials. The generic polynomial has a transitive Galois group containing an n-cycle, as the resolvent corresponding to the cyclic permutation of the roots is irreducible of degree n. It also contains 3-cycles, demonstrated by the irreducibility of the cubic resolvent polynomial whose roots are expressions like \alpha_i + \omega \alpha_j + \omega^2 \alpha_k for distinct roots \alpha_i, \alpha_j, \alpha_k and a primitive cube root of unity \omega; this resolvent has degree \binom{n}{3}, and its irreducibility implies the presence of 3-cycles, generating the alternating group A_n as the even permutations in the Galois group. To include odd permutations, consider the quadratic resolvent for transpositions, such as the polynomial whose roots are sums or products of pairs of roots (corresponding to the partition (n-2, 2)); for generic coefficients, this quadratic is irreducible, indicating that the Galois group contains a transposition (an element of order 2 swapping two roots). Since S_n is generated by an n-cycle and a transposition, the Galois group is the full S_n.^[21]^[39] Alternatively, the structure can be confirmed via the discriminant \Delta of the polynomial, given by

\Delta = \prod_{1 \leq i < j \leq n} (\alpha_i - \alpha_j)^2,

where \alpha_1, \dots, \alpha_n are the roots. For generic coefficients, \Delta is not a square in \mathbb{Q}(a_0, \dots, a_{n-1}), so the Galois group intersects the alternating group A_n properly and must be the full S_n (as the only transitive subgroups containing A_n are A_n and S_n).^[21] For the specific case of quintics (n=5), the general quintic equation has Galois group S_5, which is nonsolvable: it possesses the simple nonabelian group A_5 as a normal subgroup of index 2, and A_5 has no nontrivial normal subgroups, preventing a composition series with abelian factors. By the fundamental theorem of Galois theory, solvability by radicals requires a solvable Galois group, so the general quintic is not solvable by radicals. This argument extends directly to higher degrees, as S_n and A_n remain nonsolvable for all n \geq 5.^[21]

Specific Examples

General Quintic Equation

The general quintic equation takes the form

x^5 + a x^4 + b x^3 + c x^2 + d x + e = 0,

where a, b, c, d, e are coefficients in a field, typically the rationals \mathbb{Q}. Through the substitution x = y - \frac{a}{5}, the quartic term can be eliminated, yielding the depressed quintic y^5 + p y^3 + q y^2 + r y + s = 0, where p, q, r, s are determined by the original coefficients. The Bring-Jerrard form y^5 + t y + u = 0 can be achieved via a Tschirnhaus transformation that requires solving a quartic equation.^[40] For generic values of the coefficients, the Galois group of this depressed quintic over \mathbb{Q} is the symmetric group S_5, which is not solvable. Consequently, the roots cannot be expressed using radicals. A majority of irreducible quintics over \mathbb{Q} have Galois groups S_5 or A_5, both nonsolvable, underscoring that solvability by radicals is exceptional rather than typical.^[24]^[41] A concrete illustration is the irreducible polynomial x^5 - x - 1 = 0, whose Galois group over \mathbb{Q} is S_5. This equation has one real root, approximately 1.167, and four complex roots, none of which admit expression in terms of radicals. The Bring-Jerrard form highlights the challenge: while roots can be formally expressed using Bring radicals (hypergeometric functions solving z^5 + z + t = 0), these transcend elementary radicals, confirming the failure of radical solvability for such cases as predicted by the Abel–Ruffini theorem.^[42]^[40]^[43]

Resolvent Methods for Higher Degrees

Resolvent methods provide a framework for tackling higher-degree polynomial equations by reducing the problem to solving auxiliary polynomials whose roots encode symmetric information about the original equation's roots. For septic equations of degree 7, the Lagrange resolvent is a sextic polynomial of degree 6, constructed as the minimal polynomial over the base field for expressions of the form y = \sum_{j=1}^7 \zeta^{k j} r_j, where r_j are the roots of the septic, \zeta is a primitive 7th root of unity, and k varies to generate the roots of the resolvent.^[44] This resolvent captures the action of cyclic subgroups in the Galois group, allowing one to adjoin roots that resolve the extension step by step if the group is solvable. The construction of the resolvent R(y) facilitates expressing the septic roots in terms of the resolvent's roots combined with roots of unity, potentially using theta functions or other transcendental functions to parameterize the solutions when radicals are insufficient. For instance, in solvable cases, solving R(y) = 0 enables the original roots to be written as functions involving these transcendentials, reflecting the structure of the splitting field. However, for the general septic, the Galois group S_7 is non-solvable, so the sextic resolvent inherits this complexity and cannot be solved by radicals.^[45] Prior to applying resolvent methods, Tschirnhaus transformations are employed to simplify the septic equation by eliminating intermediate terms, reducing it to a form such as x^7 + p x + q = 0 or x^7 + a x^3 + b = 0, which preserves the invariants necessary for the resolvent construction. These transformations involve substituting y = x + c x^k + \cdots to depress the polynomial while maintaining rationality of coefficients.^[46] Despite the Abel–Ruffini theorem establishing unsolvability by radicals for degrees five and higher, resolvent methods demonstrate partial progress by confirming the non-solvability through the resolvent's Galois group while enabling solutions via elliptic modular functions or higher-genus theta functions for specific forms. For the simplified septic x^7 + a x^3 + b = 0, the sextic resolvent involves polynomial invariants like the discriminant and power sums, linking the roots to hyperelliptic curves of genus 3.^[45] This approach underscores the theorem's limitations, as it shifts the solution to more advanced transcendental tools rather than radicals.

Historical Context

Ruffini's Contributions

Paolo Ruffini (1765–1822) was an Italian mathematician and physician who studied at the University of Modena, where he graduated in medicine and mathematics before becoming a professor of mathematics and medical analysis.^[47] His early work included contributions to the theory of equations, particularly focusing on permutations of roots and their implications for solvability.^[47] In 1799, Ruffini published his seminal memoir titled Teoria generale delle equazioni, in guisa da si dimostra impossibile la soluzione algebrica delle equazioni di grado maggiore di quattro (General Theory of Equations, in which it is shown that the algebraic solution of equations of degree greater than four is impossible), claiming to prove the unsolvability of the general quintic equation by radicals.^[48] He followed this with refinements in 1803 and 1813, which provided more detailed arguments but still left some gaps. This work employed innovative arguments based on permutation groups, where he analyzed the possible ways roots could be expressed through radicals and used counting principles to derive a contradiction.^[47] A central idea was that assuming a radical solution for the quintic would imply a certain number of distinct expressions for the roots, which conflicted with the structure of permutations in the symmetric group S_5, effectively pioneering applications of permutation group concepts to algebraic solvability.^[47] Despite its pioneering nature, Ruffini's proof contained significant gaps, such as an unproven assumption of polynomial irreducibility and incomplete handling of resolvents, rendering it not fully rigorous by later standards.^[47] The memoir was largely overlooked by contemporaries like Lagrange and Legendre, receiving little attention until Cauchy referenced it in 1821.^[47] Ruffini had successfully solved specific cubic and quartic equations in prior works but shifted his focus to demonstrating the impossibility of a general radical solution for the quintic.^[47]

Abel's Proof and Developments

Niels Henrik Abel, a Norwegian mathematician born in 1802, made a groundbreaking contribution to algebra at the age of 22 by rigorously proving the unsolvability of the general quintic equation by radicals.^[49] In 1824, he self-published a memoir titled Mémoire sur les équations algébriques, où l'on démontre l'impossibilité de la résolution de l'équation générale du cinquième degré in Christiania (now Oslo), written in French.^[49] This six-page pamphlet built upon Paolo Ruffini's earlier attempt but provided a complete and correct proof, demonstrating that no general formula using radicals exists for solving quintic equations.^[50] Abel's proof introduced key innovations by linking radical solvability to properties of permutations of the roots. He showed that if an equation is solvable by radicals, then its roots can be expressed as rational functions of the roots under permutations, implying that the associated permutation group must allow for a structured descent in complexity.^[50] Drawing on Augustin-Louis Cauchy's 1815 theorem on permutations, Abel argued that such expressions would yield only a limited number of distinct values under the full symmetric group of order 120 for the quintic, leading to a contradiction since radical solutions would require far fewer variations.^[49] This descent argument reduced the problem to analyzing lower-degree resolvents, revealing an inherent incompatibility for degree five. In 1826, Abel generalized his result in a paper published in the first volume of Crelle's Journal für die reine und angewandte Mathematik, extending the unsolvability to all polynomial equations of degree five or higher.^[49] His work laid foundational ideas that influenced Évariste Galois's development of group theory, though Abel received limited recognition during his lifetime.^[49] Tragically, Abel died of tuberculosis in 1829 at the age of 26, shortly after securing a position in Berlin, and the theorem is often referred to as Abel's theorem in his honor.^[49]

Implications and Extensions

Solvable Cases

Although the Abel–Ruffini theorem establishes that general polynomial equations of degree five or higher cannot be solved by radicals, specific cases exist where such polynomials are solvable, provided their Galois groups are solvable.^[1] A polynomial is solvable by radicals if and only if the Galois group of its splitting field over the rationals is a solvable group, meaning it possesses a subnormal series with abelian factor groups.^[51] For quintic polynomials, solvability thus requires the Galois group to be a solvable transitive subgroup of the symmetric group S_5. The solvable transitive subgroups of S_5 fall into three isomorphism types: the cyclic group C_5 of order 5, the dihedral group D_5 of order 10, and the Frobenius group F_{20} (also known as the metacyclic group or affine group AGL(1,5)) of order 20.^[40] These groups have orders dividing 20, and any irreducible quintic with such a Galois group is solvable by radicals. In total, there are five isomorphism types of transitive subgroups of S_5, with the remaining two being the alternating group A_5 and S_5 itself, both nonsolvable. Examples of solvable quintics include reducible cases like x^5 - 32 = 0, whose roots are $2 \zeta^k for k = 0,1,2,3,4, where \zeta is a primitive fifth root of unity; the splitting field is \mathbb{Q}(\zeta_5), with Galois group isomorphic to (\mathbb{Z}/5\mathbb{Z})^\times \cong C_4, which is solvable via the cyclotomic extension.^[40] For irreducible examples, the trinomial x^5 + 20x + 32 = 0 has Galois group D_5 and is thus solvable by radicals.^[52] More generally, certain trinomials of the form x^5 + ax + b = 0 (the Bring–Jerrard form, to which any quintic can be reduced using Tschirnhaus transformations involving radicals) are solvable when their Galois groups are among the solvable types above, often determined by the discriminant being a perfect square or specific resolvent conditions.^[53] Quintics in Bring–Jerrard form x^5 + px + q = 0 are solvable precisely when the order of the Galois group divides 20, corresponding to the solvable transitive subgroups.^[54] However, such solvable irreducible quintics over the rationals form a set of density zero among all monic irreducible quintics with integer coefficients; specifically, for bounded height (measured by the maximum absolute value of coefficients), there are only finitely many such polynomials.^[55] To determine if a specific quintic is solvable, one computes its Galois group using methods such as evaluating resolvents (polynomials whose roots relate to invariants of the Galois action) or factoring the polynomial modulo various primes to infer the cycle index via Dedekind's theorem, which links factorization patterns to conjugacy classes in the Galois group.^[56]^[57] For instance, if factorizations modulo primes include types like a 5-cycle or double transpositions consistent only with solvable subgroups, the polynomial is solvable.

Connections to Modern Algebra

Évariste Galois's development of Galois theory in the 1830s provided the rigorous framework that completed and generalized the Abel–Ruffini theorem by establishing a precise criterion for solvability by radicals: a polynomial is solvable by radicals if and only if its Galois group is solvable.^[58] For the general polynomial of degree n \geq 5, the Galois group is the symmetric group S_n, which is not solvable, thereby validating the impossibility result through group-theoretic analysis.^[58] In computational algebra, while the theorem precludes general radical solutions, numerical and algorithmic methods address specific instances of higher-degree equations. The Lenstra–Lenstra–Lovász (LLL) lattice basis reduction algorithm enables efficient factoring of polynomials over the rationals in polynomial time, allowing decomposition into lower-degree factors that can then be solved explicitly when possible.^[59] Similarly, Felix Klein's icosahedral approach solves certain quintic equations by reducing them to the resolution of the icosahedral equation, employing elliptic modular functions rather than radicals to express roots numerically or in closed form for solvable cases.^[60] The theorem's emphasis on solvable groups has broader implications in modern algebra, where the concept of solvability parallels structures in Lie theory; solvable Lie algebras over fields of characteristic zero exhibit analogous composition series properties, facilitating the study of nilpotent and semisimple representations.^[3] In algebraic geometry, Galois theory connects to the monodromy groups of coverings over moduli spaces of curves, where the non-solvability of certain Galois groups informs the irreducibility and branching behavior of these spaces.^[3] Furthermore, the foundational role of Galois groups in finite fields underpins cryptographic protocols, such as those relying on irreducible polynomials for key generation, highlighting the theorem's indirect influence on secure systems despite the absence of direct solvability applications.^[61] Modern extensions of the theorem address unsolvability in varied contexts, including fields of positive characteristic, where the result holds analogously for separable polynomials provided the characteristic does not divide the degree, as radical extensions behave differently due to the absence of p-th roots in characteristic p.^[62] In dynamical systems, the theorem relates to the complexity of polynomial iterations, such as those defining Julia sets, where the non-solvability of higher-degree polynomial equations mirrors the intricate, non-algebraic parameterizations required to describe connected components of these sets.^[63] The Abel–Ruffini theorem inspired David Hilbert's 13th problem, posed in 1900, which queries whether seventh-degree equations can be solved via algebraic functions of at most two variables, extending the impossibility of radical solutions to broader questions of functional composition that remain partially unresolved.^[45]

References

[1]
[PDF] Abel–Ruffini theorem
Nov 3, 2012 · The Abel-Ruffini theorem states there is no general algebraic solution in radicals for polynomial equations of degree five or higher.
[2]
http://www.abelprisen.no/verker/oeuvres_1881_del1/oeuvres_completes_de_abel_nouv_ed_1_kap03_opt.pdf
[3]
The topological proof of Abel-Ruffini theorem - Project Euclid
Social approval received a proof of the analogous statement done in 1824 by N. H. Abel. Theorem (Abel and Ruffini). A general algebraic equation of degree ≥ 5.
[4]
[PDF] galois theory and the abel-ruffini theorem
This paper uses Galois Theory to prove the Abel-Ruffini Theorem, which states that polynomials of degree 5 and higher cannot be solved by radicals.
[5]
[PDF] Abel–Ruffini's Theorem: Complex but Not Complicated
Nov 5, 2020 · No formula exists for the solution to the general equation of degree five or more, using only the operations +,−,×,÷, and √ .
[6]
[PDF] Math 331-3: Abstract Algebra - Northwestern University, Lecture Notes
The fundamental notion is the following: given a field F, an extension field of F is any field E ⊇ F which contains F as a subfield. We usually say simply that ...
[7]
[PDF] Contents 2 Fields and Field Extensions - Evan Dummit
As a particular application of the basic theory of field extensions, we will be able to establish the impossibility of certain classical straightedge-and- ...
[8]
[PDF] GALOIS THEORY 2024 notes by T. D. Wooley based ... - Purdue Math
Suppose that L : M : K is a tower of field extensions and. L : K is a normal extension. ... By assumption, the field extension L : K is normal and separable.
[9]
[PDF] 1. Field extensions A subfield of a field K is a subring F ⊆ K which is ...
If the original extension K/F is also separable, then the normal closure is a Galois extension. Suppose we have finite Galois extension L/F with G = Gal(L/F).
[10]
[PDF] Summary of results: Fields and Galois Theory Alan Haynes, haynes ...
Jan 31, 2022 · We have the following result about embeddings of separable field extensions into algebraic closures. Theorem 5. Suppose that K/F is a separable ...
[11]
[PDF] Section V.9. Radical Extensions
May 1, 2018 · Definition V. 9.1. An extension field F of a field K is a radical extension of K if F = K(u1,u2,...,un) where some power of u1 lies in K and for ...
[12]
[PDF] Simple radical extensions - Keith Conrad
a, so a simple radical extension of K looks like K( n. / a), but the ... Lang, “Algebra,” 3rd revised ed., Springer-Verlag, New York, 2002. [3] L ...Missing: definition | Show results with:definition
[13]
[PDF] Galois theory lecture summary
May 3, 2018 · Definition. An extension L/K is a radical extension if and only if it can be broken down into a finite tower of simple radical extensions.
[14]
[PDF] MATH 552 NOTES – LECTURE 5 Radical field extensions
Radical field extensions: We say that a field extension E/F is radical if. E = F(u1,u2, ..., un), where some power of ui lies in F(u1, ..., ui−1) for all i.
[15]
[PDF] Solvability by radicals - Brown Math
8 dic 2013 · Theorem 7 (Galois). Let E/F be separable extension, then E/F is solvable if and only if it is solvable by radicals. Proof.
[16]
[PDF] 5.3 Solvability by Radicals - math.binghamton.edu
Apr 30, 2020 · Definition 5.7 An extension K( n pa)/K with a. 2 K and n. 1, is called a radical extension. ... the radical tower for this root extension as. K ...
[17]
Section 9.21 (09DU): Galois theory—The Stacks project
A field extension E/F is called Galois if it is algebraic, separable, and normal. It turns out that a finite extension is Galois if and only if it has the “ ...
[18]
[PDF] GALOIS THEORY 1. Automorphism groups and fixed fields Let K ...
An element x in an extension L ⊇ F is called separable over F if its minimal polynomial is separable. (If L is not normal over F, the splitting would take place ...
[19]
[PDF] LECTURE NOTES ON GALOIS THEORY 1 1.1. Splitting fields ...
d) The size of the Galois group equals the degree of the splitting field: |G(K/F)| = [K : F]. e) The action of G(K/F) on the set of roots is transitive ...
[20]
[PDF] Lecture 19 - Math 5111 (Algebra 1)
Nov 16, 2020 · Some authors refer to the automorphism group of any extension as a Galois group. We only refer to Galois groups for extensions that have the “ ...
[21]
[PDF] Fields and Galois Theory - James Milne
Theorem 3.17 (Fundamental theorem of Galois theory) Let 𝐸 be a Galois ex- ... are inverse bijections between the set of closed subgroups of 𝐺 and the set of ...<|control11|><|separator|>
[22]
[PDF] 12. The Fundamental Theorem of Galois Theory - UCSD Math
Given a subgroup H, let M = LH and given an intermediary field L/M/K, let H = Gal(L/M). Proof. This will be an easy consequence of all that has gone before.
[23]
[PDF] Galois theory lecture summary
Apr 12, 2018 · In fact, we can explicitly describe a bijection from the set of intermediate fields to the set of subgroups: F 7− → Aut(L/F). We can also ...
[24]
[PDF] Galois groups as permutation groups - Keith Conrad
Only for an irreducible polynomial does the Galois group have a special property, called transitivity, when we turn the Galois group into a subgroup of Sn. A ...
[25]
[PDF] Techniques for the Computation of Galois Groups
Instead we observe that (as f(ασ) = f(α)σ = 0σ = 0 for σ ∈ G) the Galois group acts on the roots of f. This action is transitive because f is irreducible and ...
[26]
[PDF] Galois Theory TCU Graduate Student Seminar George Gilbert ...
A finite Galois group acts transitively on a generator of the extension. Thus, the Galois group may be realized naturally as a transitive subgroup of S[L:K].
[27]
[PDF] Galois theory lecture summary
May 7, 2018 · Any group all of whose composition factors are abelian is called solvable; thus, Galois' criterion asserts that a separable f ∈ Q[x] is ...
[28]
[PDF] Lecture 24 - Math 5111 (Algebra 1)
Dec 7, 2020 · Each extension EKi+1/EKi is a simple radical extension obtained by extracting an ni th root of unity, and since all of these roots of unity are ...
[29]
Symmetric Group -- from Wolfram MathWorld
The symmetric group S_n is the group of all permutations on n symbols, and is a permutation group of order n!.
[30]
symmetric group is generated by adjacent transpositions - PlanetMath
Mar 22, 2013 · The symmetric group on {1,2,…,n} is generated by the permutations (1,2),(2,3),…,(n−1,n).
[31]
Alternating Group -- from Wolfram MathWorld
An alternating group is a group of even permutations on a set of length n , denoted A_n ; An alternating group is a normal subgroup of the permutation group, and ...
[32]
alternating group is a normal subgroup of the symmetric group
Mar 22, 2013 · Proof. |Sn|=|An||Sn/An|. | S n | = | A n | ⁢ H of G has index 2 , then H is normal in G .Missing: A_n | Show results with:A_n
[33]
Solvable Group -- from Wolfram MathWorld
A solvable group is a group having a normal series such that each normal factor is Abelian. The special case of a solvable finite group is a group whose ...
[34]
[PDF] Section VII.35. Series of Groups
Mar 20, 2014 · The group S3 is solvable because the composition series {e} <. A3 < S3 has factor groups A3/{e}∼= A3. ∼. = Z3 and S3/A3. ∼. = Z2, both of which ...
[35]
[PDF] 4.2 VII-35. Series of Groups
The group S3 is solvable since it has a composition series. {e} < A3 < S3 where A3/{e} ≃ Z3 and S3/A3 ≃ Z2 are abelian. 2. (opt: The Ascending Central Series) ...
[36]
[PDF] SIMPLICITY OF An - KEITH CONRAD
When n ≥ 3 the group Sn is not simple since it has the normal subgroup An of index 2. Theorem 1.1. For n ≥ 5, the group An is simple. This is due to Camille ...<|separator|>
[37]
Simple Group -- from Wolfram MathWorld
A simple group is a group whose only normal subgroups are the trivial subgroup of order one and the improper subgroup consisting of the entire original group.
[38]
Commutator Subgroup -- from Wolfram MathWorld
. The commutator subgroup of the symmetric group is the alternating group. The commutator subgroup of the alternating group A_n is the whole group A_n ...<|separator|>
[39]
[PDF] RECOGNIZING GALOIS GROUPS Sn AND An - Keith Conrad
So the hypotheses of Theorem 2.1 and 2.2 are satisfied by Sn for n ≥ 2 and An for n ≥ 3: they are transitive subgroups of themselves and have a p-cycle for some ...
[40]
Quintic Equation -- from Wolfram MathWorld
... symmetric group S_n, metacyclic group M_n ... Solvability of a quintic is then predicated by its corresponding group being a solvable group.
[41]
[PDF] Galois Theory and the Insolvability of the Quintic
May 22, 2022 · The splitting field of X5 − 6X + 3 has Galois group S5... Now we give an example of a quintic equation whose roots are not expressible by ...
[42]
[PDF] The Galois group of x n - x - 1 over Q - Keith Conrad
Step 1: The Galois group of the splitting field of xn − x − 1 over Q embeds into Sn by acting on the roots of xn − x − 1 and fixing a labeling of the roots.
[43]
[PDF] arXiv:2112.06021v2 [math.GM] 29 Dec 2021
Dec 29, 2021 · The simplest form of quintic equation that is unsolvable in radicals is Bring-Jerrard quintic equation, x5 + Ax + B = 0.<|control11|><|separator|>
[44]
[PDF] Computing the Lagrange resolvent by effectiveness of Galois Theorem
In this article, we introduce a new method to calculate Lagrange resolvent. This technique is based on Lagrange's algorithm and it enables to calculate ...
[45]
[PDF] Resolvent degree, Hilbert's 13th Problem and geometry
Abstract. We develop the theory of resolvent degree, introduced by Brauer [Brau2] in order to study the complexity of formulas for roots of polynomials and ...
[46]
[PDF] Polynomial Transformations of Tschirnhaus, Bring and Jerrard
Tschirnhaus demonstrated the utility of his transformation by apparently solving the cubic equation in a way different from Cardano.
[47]
Paolo Ruffini (1765 - 1822) - Biography - MacTutor
Paolo Ruffini was an Italian mathematician who gave a proof that the quintic equation could not be solved earlier than Abel. Thumbnail of Paolo Ruffini View ...
[48]
Mathematical Treasure: Ruffini's Theory of Equations
He was the first mathematician to state and prove that an algebraic equation of degree greater than four cannot be solved by radicals.
[49]
[PDF] Abel Answers the Question of the Quintic - IMA
Feb 20, 2025 · ian mathematician Paolo Ruffini (1765–1822) published a book- length proof that the general quintic equation could not be solved by radicals ...
[50]
[PDF] Abel and the insolvability of the quintic - Mathematics
In this paper we will treat Abel's proof that a general quintic is not solvable by radicals, a startling proof when most mathematicians of the day thought it ...
[51]
https://kconrad.math.uconn.edu/blurbs/galoistheory/cubicquartic.pdf
[52]
[PDF] An Algorithm for Solving Solvable Polynomial Equations of ... - arXiv
Mar 26, 2022 · 4 One Simple Example: x5 + 20x +32=0. The Galois group G of f(x) = x5 +20x+32 is the dihedral group D5 [9], and the generators are (1,2,3,4,5) ...
[53]
[PDF] Characterization of Solvable Quintics
Hence the equation x5 +. 2px + 2p² 0 is not solvable by radicals. Other examples of solvable quintics are given below together with their Galois groups. €. 1 ...
[54]
[PDF] On Irreducible Rational Quintics - Ryan Kavanagh
We present in section 4 a method by Spearman and. Williams [SW94] for solving solvable quintics in Bring-Jerrard form by radi- cals, and we conclude by applying ...
[55]
[PDF] arXiv:2106.14593v2 [math.NT] 30 Jan 2023
Jan 30, 2023 · How often is a quintic polynomial solvable by radicals? We establish that the number of such polynomials, monic and irreducible with integer ...
[56]
[PDF] Galois groups over Q and factorizations mod p - Keith Conrad
The following striking theorem of Dedekind tells us cycle types of elements of Gal(K/Q) as a permutation of the roots of f(T) by how f(T) factors modulo primes.
[57]
[PDF] Characterization and computation of the Galois group of ...
The so-called resolvent method computes Galf once a classification of the transitive subgroups of Sn is known. Step 1: Find invariant polynomials. For each ...
[58]
[PDF] an introduction to galois theory and the abel-ruffini theorem
The Abel-Ruffini Theorem proves that there is no explicit formula using our basic operations to find the roots of a polynomial with a degree of 5 or greater.
[59]
[PDF] Solving problems with the LLL algorithm - FSU Math
Factoring in Q[x] can be done in polynomial time. Proof sketch: Compute a root of f to precision a. Use the previous slide to compute its minpoly.
[60]
[1308.0955] On Klein's Icosahedral Solution of the Quintic - arXiv
Aug 5, 2013 · We present an exposition of the icosahedral solution of the quintic equation first described in Klein's classic work "Lectures on the icosahedron and the ...
[61]
[PDF] Part II - Galois Theory - Dexter Chua
Galois theory studies field extensions, and is often said to be the study of field extensions. It grew from the desire to solve polynomial equations.
[62]
[PDF] Unsolvability of the Quintic Formalized in Dependent Type Theory
May 2, 2021 · corresponding to our definition of “radical extension” and the other corresponding to our ... 15. Serge Lang. Algebra. Graduate Texts in ...
[63]
Local connectivity of the Julia set of real polynomials - math - arXiv
Apr 26, 1995 · In this paper we shall prove the following Main Theorem: Let f be a polynomial of the form f(z)=z^d +c with d an even integer and c real.Missing: Abel- Ruffini