Galois & Abel
Tragedy, genius, and the birth of abstract algebra
16.1 The Quintic Problem
Since antiquity, mathematicians had sought formulas for solving polynomial equations. The Babylonians solved quadratics around 2000 BCE. In the 16th century, Cardano and his contemporaries found formulas for cubics and quartics. The natural question was: can the general quintic equation
be solved by a formula involving only addition, subtraction, multiplication, division, and the extraction of$n$th roots (i.e., by radicals)? For over 250 years, the world's greatest mathematicians — including Euler, Lagrange, and many others — attempted to find such a formula. All failed.
Lagrange made a crucial observation around 1770: the known solution methods for cubics and quartics all worked by finding resolvent equations of lower degree. For the cubic, one solves an auxiliary quadratic; for the quartic, an auxiliary cubic. But for the quintic, any resolvent seemed to produce an equation of degree six or higher. Lagrange suspected that the quintic was fundamentally different, but he could not prove it.
Lagrange's analysis went further. He studied the role of permutations of roots in the solution process. He observed that the resolvent formulas for quadratics, cubics, and quartics all involve rational functions of the roots that take fewer values under permutation than one might expect. For a polynomial of degree $n$, the roots can be permuted in $n!$ ways, but the key resolvent expressions are invariant under many of these permutations. This reduction in the number of distinct values is precisely what allows the resolvent to have lower degree. For the quintic, Lagrange could find no such reduction — a hint that something deep was at work.
The resolution of this 250-year quest would come from two young mathematicians who both died tragically young: Niels Henrik Abel and Évariste Galois.
16.2 Niels Henrik Abel (1802 – 1829)
Niels Henrik Abel was born on August 5, 1802 in Finnøy, Norway, to a poor family. His father was a Protestant minister who fell into alcoholism and died young, leaving the family destitute. Despite these hardships, Abel's mathematical genius was recognized early by his teacher Bernt Michael Holmboe, who introduced him to the works of Euler, Lagrange, and Gauss.
Abel's Short Life
- 1802 — Born in Finnøy, Norway
- 1820 — Father dies; family plunges into poverty
- 1821 — Enters the University of Christiania (Oslo)
- 1823 — Believes he has solved the quintic; discovers his own error
- 1824 — Publishes proof of the impossibility of solving the quintic by radicals
- 1825–1827 — Travels through Europe; meets mathematicians in Berlin and Paris
- 1826 — Submits his masterwork on elliptic functions to the French Academy; Cauchy and Legendre neglect it
- 1829 — Dies of tuberculosis at age 26, days before a letter offering him a professorship arrives
Abel's journey to the impossibility theorem had a false start. In 1821, as a nineteen-year-old student, he believed he had found a general formula for solving the quintic. He sent his work to the Danish mathematician Ferdinand Degen, who asked him to test the formula on specific examples. Upon doing so, Abel discovered his error. This failure, however, led him to consider the opposite question: perhaps no such formula exists at all. Within two years, he had his proof.
16.3 The Abel–Ruffini Theorem
Paolo Ruffini had published an incomplete proof of the quintic's insolvability in 1799 in a 500-page treatise. Ruffini's argument contained the essential ideas but had significant gaps — he assumed without proof that any radical expression for the roots must have a particular form. The mathematical community was skeptical, and Ruffini spent the last years of his life trying to convince his peers. Cauchy was one of the few who acknowledged Ruffini's contribution, writing to him: “Your memoir on the general resolution of equations is a work I have always regarded as deserving the attention of mathematicians.”
Abel's Impossibility Theorem (Abel–Ruffini Theorem, 1824)
There is no general algebraic formula — expressible in terms of radicals — for the roots of polynomial equations of degree five or higher.
Sketch of Abel's proof:
Step 1. Suppose a root of the general quintic can be written as a formula involving the coefficients and nested radicals. Any such expression defines a tower of field extensions:
where each $F_{i+1} = F_i(\alpha_i)$ with $\alpha_i^{p_i} \in F_i$ for some prime $p_i$. Each step adjoins a $p_i$th root of an element already in the current field.
Step 2. Abel showed that any such radical expression for the roots must, when expanded as a function of the five roots $x_1, \ldots, x_5$, take at most $p_i$ distinct values under permutations of the roots at each level of the tower.
Step 3. He proved a key lemma: if a rational function of five quantities takes fewer than five distinct values under all permutations, then it takes either 1 or 2 values. A function of five variables taking exactly $p$ values under all 120 permutations forces $p$ to divide $120 = 5!$, but additional constraints from the tower structure rule out all primes $p \geq 3$ at the critical steps.
Step 4. This leads to a contradiction: the tower cannot be long enough to express the general root, because the group of permutations of five objects cannot be decomposed into the required chain of cyclic extensions.
Abel published his proof as a pamphlet at his own expense, compressing it to just six pages to save on printing costs. Unfortunately, this brevity made it difficult to read, and it initially received little attention. When Abel sent it to Gauss, the Prince of Mathematicians reportedly set it aside without reading it, assuming it was yet another crank's claimed quintic solution. It was not until Abel traveled to Berlin and met August Leopold Crelle — who had just founded his influential mathematics journal — that his work began to gain recognition.
16.4 Abel's Other Contributions
Despite his tragically short life, Abel made contributions that shaped multiple areas of mathematics.
Abelian Groups
A group $G$ is called abelian (or commutative) if$ab = ba$ for all $a, b \in G$. The term honours Abel's work on commutative structures in his investigations of solvability. Examples include$(\mathbb{Z}, +)$, $(\mathbb{Q}^*, \times)$, and the group of points on an elliptic curve.
Abel's theorem on elliptic integrals generalized Euler's addition theorem for elliptic integrals to a vast class of algebraic integrals. Given an algebraic curve $f(x, y) = 0$, Abel showed that certain sums of integrals
can be expressed in terms of a small number of “fundamental” integrals. This was later recognized as one of the deepest results in algebraic geometry. Jacobi called it “the greatest discovery of our time,” and it opened the door to the theory of Riemann surfaces and algebraic curves.
Abel's Convergence Test
If $\sum a_n$ converges and $\{b_n\}$ is a monotone bounded sequence, then$\sum a_n b_n$ converges.
More generally, Abel summation (summation by parts) provides the discrete analogue of integration by parts:
where $A_k = \sum_{j=0}^{k} a_j$ is the partial sum.
16.5 Évariste Galois (1811 – 1832)
Évariste Galois lived one of the most dramatic lives in the history of science. Born on October 25, 1811 near Paris, he was the son of a liberal politician who became mayor of Bourg-la-Reine. Galois showed little interest in mathematics until age fifteen, when he discovered the works of Legendre and Lagrange — which he reportedly read “like a novel.”
Galois's Turbulent Life
- 1811 — Born near Paris
- 1827 — Discovers Legendre and Lagrange; begins mathematical research
- 1828 — Fails entrance exam to the École Polytechnique (twice)
- 1829 — Father commits suicide after a political scandal; Galois submits first paper to Cauchy, who loses it
- 1830 — Submits paper to Fourier, who dies before reading it; participates in the July Revolution
- 1831 — Arrested twice for political activism; submits final paper to Poisson, who calls it “incomprehensible”
- May 29, 1832 — The night before his duel, writes his famous letter summarizing his mathematical ideas
- May 30, 1832 — Shot in a duel; dies the next morning at age 20
Galois's mathematical career was marked by astonishing genius and equally astonishing bad luck. He twice failed the entrance examination to the prestigious École Polytechnique — legend has it that he threw an eraser at the examiner in frustration. He submitted his groundbreaking papers to the French Academy three times: Cauchy lost the first; Fourier died before reading the second; Poisson rejected the third as “not sufficiently developed.”
A fervent republican during the turbulent years following the French Revolution of 1830, Galois was deeply involved in radical politics. He joined the Society of the Friends of the People and the Artillery of the National Guard, both republican organizations. At a banquet in May 1831, he allegedly raised a glass and a dagger, appearing to toast to the king's death. He was arrested, tried, and acquitted — only to be arrested again weeks later for wearing the banned uniform of the National Guard during a Bastille Day demonstration. He spent months in Sainte-Pélagie prison, where he continued his mathematical work and, according to some accounts, attempted suicide in despair.
The circumstances surrounding Galois's fatal duel remain shrouded in mystery. It likely involved a romantic entanglement — a woman named Stéphanie-Félicie Poterin du Motel appears in his papers, her name scratched out and replaced with despairing comments. Some historians have speculated that he was provoked into the duel by political enemies; others believe it was a genuine affair of honor. What is certain is that on the evening of May 29, 1832, knowing he might die the next morning, Galois sat down to write.
16.6 The Letter of May 29, 1832
The night before the duel, Galois stayed up writing a long letter to his friend Auguste Chevalier, frantically summarizing his mathematical discoveries. This letter is one of the most extraordinary documents in the history of science. In it, Galois outlined the foundations of what we now call Galois theory, described the connection between field extensions and groups, and stated results that would take other mathematicians decades to fully understand and prove.
“I have made some new discoveries in analysis. The first concerns the theory of equations; the others concern integral functions. In the theory of equations, I have investigated the conditions for the solvability of equations by radicals; this has given me the opportunity to deepen this theory and describe all the transformations possible on an equation even when it is not solvable by radicals.” — Évariste Galois, letter to Auguste Chevalier
In the margins, Galois scribbled desperate annotations: “I have not time; I have not time” and “There is something to complete in this demonstration. I do not have the time.” He asked Chevalier to publish the letter and to ask Jacobi and Gauss to give their opinion “not on the truth but on the importance of these theorems.”
Galois was shot in the abdomen the next morning and was left where he fell. A peasant found him and brought him to a hospital, where his younger brother Alfred rushed to his side. According to Alfred, Galois's last words were: “Do not cry, Alfred. I need all my courage to die at twenty.” He died on May 31, 1832.
His mathematical manuscripts, when finally studied by Joseph Liouville in 1843, were recognized as containing one of the most profound advances in the entire history of mathematics. Liouville announced to the Academy: “I have experienced an intense pleasure at the moment when, having filled in some minor gaps, I saw the complete correctness of the method by which Galois proves, in particular, this beautiful theorem.”
16.7 Permutation Groups
At the heart of Galois theory lies the concept of a permutation group. A permutation of a set is a bijection from the set to itself, and the collection of all permutations forms a group under composition.
The Symmetric Group
The symmetric group $S_n$ is the group of all permutations of$\{1, 2, \ldots, n\}$. It has order $n!$. Elements can be written in cycle notation: for example, in $S_4$ the permutation that sends $1 \to 2, 2 \to 3, 3 \to 1, 4 \to 4$ is written as $(1\;2\;3)$.
Elements of S_3
The group $S_3$ has $3! = 6$ elements:
- $e = ()$ — the identity
- $(1\;2)$, $(1\;3)$, $(2\;3)$ — the three transpositions
- $(1\;2\;3)$, $(1\;3\;2)$ — the two 3-cycles
The subgroup $A_3 = \{e, (1\;2\;3), (1\;3\;2)\} \cong \mathbb{Z}/3\mathbb{Z}$ consists of the even permutations (those that can be written as a product of an even number of transpositions).
Normal Subgroups
A subgroup $N$ of a group $G$ is normal (written$N \trianglelefteq G$) if $gNg^{-1} = N$ for every $g \in G$. Equivalently, the left and right cosets of $N$ coincide. Normal subgroups are precisely the subgroups for which the quotient $G/N$ can be given a natural group structure.
The Alternating Group
The alternating group $A_n$ is the subgroup of $S_n$consisting of all even permutations. It has order $n!/2$ and is always a normal subgroup of $S_n$ (being the kernel of the sign homomorphism). For $n \geq 5$,$A_n$ is a simple group — it has no proper normal subgroups. This fact is the key to the insolvability of the quintic.
16.8 Galois Theory
Galois's fundamental insight was to associate a group of symmetries to every polynomial equation, and to show that the solvability of the equation by radicals is equivalent to a group-theoretic property of this symmetry group.
The Galois Group
Let $f(x) \in F[x]$ be a polynomial with coefficients in a field $F$, and let$E$ be its splitting field — the smallest field extension of $F$containing all roots of $f$.
The Galois group $\text{Gal}(E/F)$ is the group of all field automorphisms of $E$ that fix every element of $F$. These automorphisms permute the roots of$f$ among themselves, so the Galois group naturally embeds as a subgroup of $S_n$where $n = \deg f$.
Solvable Groups
A group $G$ is solvable if there exists a chain of subgroups:
such that each quotient $G_{i+1}/G_i$ is abelian (equivalently, cyclic of prime order). The symbol $\trianglelefteq$ denotes a normal subgroup. The name “solvable” comes directly from the connection to solvability of equations by radicals.
Galois's Fundamental Theorem
A polynomial equation $f(x) = 0$ with coefficients in $F$ is solvable by radicals if and only if its Galois group $\text{Gal}(E/F)$ is a solvable group.
The Galois correspondence establishes a bijection between intermediate fields $F \subseteq K \subseteq E$and subgroups of $\text{Gal}(E/F)$:
Moreover, $K/F$ is a normal extension if and only if $H$ is a normal subgroup of $\text{Gal}(E/F)$, and in that case $\text{Gal}(K/F) \cong \text{Gal}(E/F)/H$.
This is one of the great unifying ideas in mathematics: it translates questions about field extensions (algebra) into questions about group structure (symmetry), where powerful tools are available. The lattice of subfields is “flipped upside down” compared to the lattice of subgroups — larger fields correspond to smaller subgroups and vice versa.
16.9 Galois Groups of Specific Polynomials
To develop intuition, let us compute the Galois groups of several concrete polynomials.
The Quadratic: x^2 - 2
The roots are $\pm\sqrt{2}$. The splitting field is $\mathbb{Q}(\sqrt{2})$with $[\mathbb{Q}(\sqrt{2}) : \mathbb{Q}] = 2$. The only non-trivial automorphism sends $\sqrt{2} \mapsto -\sqrt{2}$. Thus:
This is abelian, hence solvable — consistent with the quadratic formula.
The Cubic: x^3 - 2
The three roots are:
where $\omega = e^{2\pi i/3} = \frac{-1 + i\sqrt{3}}{2}$ is a primitive cube root of unity. The splitting field is $E = \mathbb{Q}(\sqrt[3]{2}, \omega)$ with$[E : \mathbb{Q}] = 6$. Since $|S_3| = 6$ and the Galois group embeds into $S_3$:
The group $S_3$ has the composition series$\{e\} \trianglelefteq A_3 \trianglelefteq S_3$ with abelian quotients, so it is solvable — confirming Cardano's formula works.
The Quartic: x^4 - 2
The four roots are $\sqrt[4]{2}, \; i\sqrt[4]{2}, \; -\sqrt[4]{2}, \; -i\sqrt[4]{2}$. The splitting field is $\mathbb{Q}(\sqrt[4]{2}, i)$ with degree 8 over$\mathbb{Q}$. The Galois group is the dihedral group$D_4$ of order 8. The composition series is:
All quotients are abelian, so $D_4$ is solvable.
A Cyclotomic Polynomial: x^5 - 1
The roots are the 5th roots of unity: $1, \omega, \omega^2, \omega^3, \omega^4$where $\omega = e^{2\pi i/5}$. The minimal polynomial of $\omega$over $\mathbb{Q}$ is the cyclotomic polynomial$\Phi_5(x) = x^4 + x^3 + x^2 + x + 1$. The Galois group is:
This is cyclic, hence solvable. The roots can indeed be expressed using radicals involving $\sqrt{5}$ and nested square roots.
16.10 Why the Quintic Fails: S_5 Is Not Solvable
The Galois group of the general quintic over $\mathbb{Q}(a, b, c, d, e)$ is$S_5$, the symmetric group on 5 elements, with $|S_5| = 120$. The key theorem is:
S_5 is Not Solvable
The symmetric group $S_5$ is not solvable. This is because its only proper normal subgroup is the alternating group $A_5$, and $A_5$ is simple (has no proper normal subgroups at all).
Proof that $A_5$ is simple:
$|A_5| = 60$. The conjugacy classes in $A_5$ have sizes 1, 12, 12, 15, 20 (corresponding to the identity, two classes of 5-cycles, products of two disjoint transpositions, and 3-cycles respectively).
Any normal subgroup must be a union of conjugacy classes including the identity. The possible orders of such unions (including 1) are: 1, 13, 13, 16, 21, 25, 25, 28, 28, 33, 33, 36, 40, 41, 45, 48, 52, 53, 57, 60. But by Lagrange's theorem, the order of a subgroup must divide$60$. The divisors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60. None of the possible union sizes (other than 1 and 60) is a divisor of 60. Therefore $A_5$ has no proper normal subgroups.
Since the only composition series for $S_5$ is:
and the quotient $A_5/\{e\} \cong A_5$ is not abelian (it is a non-abelian simple group of order 60), $S_5$ is not solvable. The composition series cannot be refined further because $A_5$ is simple.
By Galois's fundamental theorem, since the Galois group $S_5$ is not solvable, the general quintic equation is not solvable by radicals. Specific quintics may have solvable Galois groups (and hence be solvable by radicals), but there is no universal formula that works for all quintics.
It is instructive to contrast the situation with lower degrees. For $n \leq 4$, the symmetric group $S_n$ is solvable:
- $S_1 = \{e\}$ — trivially solvable
- $S_2 \cong \mathbb{Z}/2\mathbb{Z}$ — abelian, hence solvable
- $S_3$: composition series $\{e\} \trianglelefteq A_3 \trianglelefteq S_3$ with abelian quotients
- $S_4$: composition series $\{e\} \trianglelefteq V_4 \trianglelefteq A_4 \trianglelefteq S_4$ where $V_4$ is the Klein four-group — all quotients are abelian
At $n = 5$, the alternating group $A_5$ becomes simple and non-abelian, breaking the pattern. For all $n \geq 5$, $A_n$ is simple and non-abelian, so $S_n$ is not solvable. This is why degree 5 is the precise threshold where the general formula fails.
A Specific Unsolvable Quintic
The polynomial $f(x) = x^5 - 4x + 2$ over $\mathbb{Q}$ is irreducible (by Eisenstein's criterion at $p = 2$) and has Galois group $S_5$. To verify this:
- Since $f$ is irreducible of degree 5 over $\mathbb{Q}$, its Galois group contains a 5-cycle (any irreducible polynomial of prime degree has this property).
- By analyzing the derivative $f'(x) = 5x^4 - 4$, one finds that $f$ has exactly three real roots and one pair of complex conjugate roots.
- Complex conjugation acts as a transposition on the roots.
- A subgroup of $S_5$ containing both a 5-cycle and a transposition must be all of $S_5$.
Therefore $\text{Gal}(f) \cong S_5$, and $f$ is not solvable by radicals.
A Solvable Quintic
The polynomial $x^5 - 1 = 0$ has roots $1, \omega, \omega^2, \omega^3, \omega^4$where $\omega = e^{2\pi i/5}$. Its Galois group over $\mathbb{Q}$ is$\mathbb{Z}/4\mathbb{Z}$ (cyclic, hence solvable). The roots can be expressed using radicals:
This expression involves only square roots — reflecting the fact that $\mathbb{Z}/4\mathbb{Z}$has a composition series with quotients of order 2.
16.11 Groups Beyond Algebra
The concept of a group, born from Galois's work on polynomial equations, turned out to be one of the most universal structures in all of mathematics and science. The modern axiomatic definition crystallized in the late 19th century:
Group Axioms
A group is a set $G$ equipped with a binary operation $\cdot$ satisfying:
- Closure: For all $a, b \in G$, $a \cdot b \in G$
- Associativity: $(a \cdot b) \cdot c = a \cdot (b \cdot c)$ for all $a, b, c \in G$
- Identity: There exists $e \in G$ such that $e \cdot a = a \cdot e = a$ for all $a$
- Inverse: For each $a \in G$, there exists $a^{-1} \in G$ with $a \cdot a^{-1} = a^{-1} \cdot a = e$
Groups appear everywhere:
- Symmetry groups: The rotations and reflections of a geometric figure (dihedral groups, the rotation group SO(3))
- Crystallography: The 230 space groups classify all possible crystal symmetries in 3D
- Physics: The Lorentz group (special relativity), gauge groups SU(3) × SU(2) × U(1) (the Standard Model of particle physics)
- Number theory: Class groups, unit groups, Galois groups
- Topology: The fundamental group classifies loops in a space up to continuous deformation
- Coding theory: Error-correcting codes use group structure for efficient encoding and decoding
16.12 Augustin-Louis Cauchy — The Rigorization of Analysis
Augustin-Louis Cauchy (1789–1857) was the mathematician most responsible for putting analysis on a rigorous foundation. Before Cauchy, concepts like limit, continuity, derivative, and integral were understood intuitively but lacked precise definitions. Cauchy provided those definitions, transforming analysis from an art into a science.
Cauchy's Epsilon-Delta Definitions
Limit: $\lim_{x \to a} f(x) = L$ means: for every $\varepsilon > 0$, there exists $\delta > 0$ such that $|f(x) - L| < \varepsilon$ whenever$0 < |x - a| < \delta$.
Continuity: $f$ is continuous at $a$ if$\lim_{x \to a} f(x) = f(a)$.
Convergence: A sequence $\{a_n\}$ converges to $L$ if for every$\varepsilon > 0$, there exists $N$ such that $|a_n - L| < \varepsilon$for all $n > N$.
Cauchy Sequences
A sequence $\{a_n\}$ is a Cauchy sequence if for every $\varepsilon > 0$, there exists $N$ such that $|a_m - a_n| < \varepsilon$ for all $m, n > N$.
In $\mathbb{R}$, a sequence converges if and only if it is Cauchy. This property (completeness) is what distinguishes $\mathbb{R}$ from $\mathbb{Q}$.
Cauchy's contributions to complex analysis were equally foundational:
Cauchy's Integral Theorem
If $f(z)$ is analytic (holomorphic) on and inside a simple closed contour $C$, then:
Cauchy's Integral Formula
If $f$ is analytic inside and on $C$, and $a$ is a point inside $C$, then:
and more generally, $f^{(n)}(a) = \frac{n!}{2\pi i}\oint_C \frac{f(z)}{(z-a)^{n+1}}\,dz$.
The Residue Theorem
If $f$ is analytic inside $C$ except at isolated singularities $z_1, \ldots, z_n$, then:
where $\text{Res}(f, z_k)$ is the residue of $f$ at $z_k$ (the coefficient of $(z - z_k)^{-1}$ in the Laurent series).
These theorems are among the most powerful results in all of analysis. The residue theorem alone provides elegant methods for evaluating definite integrals, summing series, and proving results in number theory that seem to have nothing to do with complex numbers.
16.13 Legacy — Algebra Becomes Structural
The work of Galois and Abel marks one of the great turning points in the history of mathematics. Before them, algebra was primarily concerned with solving equations — finding numerical values of unknowns. After them, algebra became the study of abstract structures: groups, rings, fields, and the relationships between them.
This shift from computation to structure was enormously powerful. Galois theory showed that the structure of a group — its subgroups, quotients, and composition series — reveals deep truths about objects that seem unrelated to groups at all. This principle of studying structures rather than individual objects became the guiding philosophy of modern mathematics.
The influence of Galois theory extends far beyond its original context. It inspired the classification of finite simple groups — one of the largest collaborative projects in mathematical history, completed in 2004 after decades of work by hundreds of mathematicians. It provided the framework for understanding why certain geometric constructions (trisecting an angle, doubling the cube) are impossible with compass and straightedge alone. And it laid the conceptual foundation for modern algebraic number theory, algebraic geometry, and even parts of theoretical physics.
The tragedy of Abel and Galois — both dying before their work was understood — remains a poignant reminder of how fragile the progress of human knowledge can be. Abel was 26 and Galois was 20. One can only imagine what they might have accomplished with full lifetimes. The Norwegian mathematician Hermite wrote: “Abel has left mathematicians enough to keep them busy for five hundred years.” The same might be said of Galois.
Today, the Abel Prize — established in 2003 by the Norwegian government — is one of the highest honors in mathematics, recognizing contributions of extraordinary depth and influence. It stands as a fitting tribute to a young man who died in poverty, unknown, but whose ideas transformed mathematics forever.