Évariste Galois

Évariste Galois (1811–1832) was a French mathematician who, before dying in a duel at age 20, founded what is now called Galois theory — the deep connection between field extensions and group theory that explains why the general polynomial equation of degree five or higher cannot be solved by radicals. His key result: a polynomial is solvable by radicals if and only if its associated symmetry group (the Galois group) is a solvable group. His work launched modern abstract algebra.

Évariste Galois discovered that the solvability of an algebraic equation by radicals is governed by the structure of a finite group attached to the equation. Where his predecessors (Lagrange, Ruffini, Abel) asked whether the quintic could be solved, Galois explained why, replacing computational struggle with a structural criterion. His central insight was to associate to each polynomial a group of permutations of its roots that preserve all algebraic relations among them; the equation is solvable by radicals exactly when this group can be broken down into a chain of abelian pieces. The surprising fact is that nearly all of this was written, in barely legible form, by a self-taught teenager twice rejected by the École Polytechnique, and lost or dismissed by the leading mathematicians of his day — only recognized 14 years after his death when Liouville published his papers.

The Core Insight

Symmetry controls solvability. To each polynomial equation, attach the group of all permutations of its roots that respect every polynomial relation with rational coefficients. Algebraic operations on the equation correspond to operations on this group. Solving "by radicals" (using

+, -, \times, \div

and

n

th roots) translates into building the group from successive abelian layers. Hence: solvable equation ⟺ solvable group. Because the symmetric group

S_{5}

is not solvable, the general quintic is not solvable by radicals.

Formal Statement

Field Extension and Galois Group

Let $K \subseteq L$ be a field extension. The Galois group $Gal (L / K)$ is the group of all field automorphisms of $L$ that fix every element of $K$ :

Gal (L / K) = {σ : L \to L ∣ σ is a field automorphism, σ (a) = a \forall a \in K}

⟳An automorphism must preserve

+

and

\times

; fixing

K

forces it to permute only the "new" elements, in particular the roots of any polynomial over

K

Here $K$ is the base field, $L$ is the extension field, and $σ$ ranges over the structure-preserving bijections of $L$ fixing $K$ pointwise. When $L$ is the splitting field of a separable polynomial over $K$ , the extension is called Galois, and we write $Gal (L / K)$ for its Galois group.

Solvable Group

A finite group $G$ is solvable if there exists a chain of subgroups (a subnormal series)

{e} = G_{0} ◃ G_{1} ◃ G_{2} ◃ \dots ◃ G_{n} = G

⟳Each

G_{i}

is normal in

G_{i + 1}

(the symbol

◃

), so the quotient

G_{i + 1} / G_{i}

is a well-defined group.

such that every quotient $G_{i + 1} / G_{i}$ is abelian (commutative). The integer $n$ measures how many abelian "layers" build the group. Every abelian group is solvable ( $n = 1$ ); $S_{2}, S_{3}, S_{4}$ are solvable, but $S_{5}$ is not.

The Fundamental Theorem of Galois Theory

Let $L / K$ be a finite Galois extension with group $G = Gal (L / K)$ . There is an inclusion-reversing bijection between intermediate fields $K \subseteq M \subseteq L$ and subgroups $H \leq G$ , given by

M ⟼ Gal (L / M), H ⟼ L^{H} = {x \in L ∣ σ (x) = x \forall σ \in H}

⟳Larger subgroups fix smaller subfields; the fixed-field operator and the Galois-group operator are mutually inverse on a Galois extension (Artin's theorem on linear independence of characters guarantees no collapse).

Moreover $[L : M] = ∣ Gal (L / M) ∣$ and $[M : K] = [G : Gal (L / M)]$ , and $M / K$ is Galois iff $Gal (L / M)$ is normal in $G$ , in which case $Gal (M / K) ≅ G / Gal (L / M)$ .

Galois's Solvability Criterion

A polynomial $f (x) \in K$ (with $K$ of characteristic $0$ ) is solvable by radicals if and only if its Galois group $Gal (L / K)$ — where $L$ is the splitting field of $f$ — is a solvable group.

f solvable by radicals ⟺ Gal (L / K) is solvable

⟳A radical extension corresponds to adjoining roots of

x^{n} - a

, whose Galois group (over a field with roots of unity) is cyclic, hence abelian; chaining radicals builds a solvable group, and conversely.

Why It Holds

Sketch: Why $S_5$ Being Non-Solvable Kills the General Quintic

Step 1. The general polynomial of degree

n

over the field

K = Q (a_{1}, \dots, a_{n})

of indeterminate coefficients has Galois group equal to the full symmetric group

S_{n}

, because no nontrivial polynomial relation constrains the roots beyond the elementary symmetric ones. Step 2. By the Solvability Criterion [4], the general degree-

n

equation is solvable by radicals iff

S_{n}

is a solvable group. Step 3. Examine the derived series. For

n \leq 4

S_{n}

is solvable:

S_{4} ▹ A_{4} ▹ V_{4} ▹ {e}, quotients Z_{2}, Z_{3}, Z_{2} \times Z_{2}

⟳

A_{4} / V_{4} ≅ Z_{3}

and

V_{4}

(the Klein four-group) is abelian; all quotients are abelian, so the series witnesses solvability.

Step 4. For

n \geq 5

, the alternating group

A_{n}

is simple and non-abelian. In particular

A_{5}

(order 60) has no nontrivial normal subgroups, so the only subnormal series of

S_{5}

S_{5} ▹ A_{5} ▹ {e}

, whose quotient

A_{5} / {e} = A_{5}

is non-abelian. Step 5. Therefore

S_{5}

is not solvable, so by [4] the general quintic is not solvable by radicals.

■

:::svg[The Galois correspondence: subgroups of $S_{3}$ versus subfields of $Q (32, ω)$ ]

Évariste Galois | 1811–1832

Born in Bourg-la-Reine near Paris to a politically liberal mayor father, Galois absorbed mathematics largely on his own, reading Legendre and Lagrange in the original. He twice failed the entrance exam to the École Polytechnique — once reportedly throwing an eraser at an examiner — and was expelled from the École Normale for a political letter. A passionate Republican during the turmoil following the 1830 Revolution, he was imprisoned twice. On 30 May 1832 he died of wounds from a duel whose cause remains obscure (a romantic entanglement, possibly engineered politically). The night before, he frantically annotated his manuscripts, writing "Je n'ai pas le temps" ("I have no time") in the margins. His complete mathematical output fills only about sixty pages, yet it reoriented all of algebra.

Building Intuition

Think of the roots of a polynomial as objects you can shuffle. Some shuffles are "invisible" from the rational numbers' point of view — if $2$ and $- 2$ are roots, swapping them changes nothing a rational observer could detect, since both satisfy $x^{2} - 2 = 0$ . The collection of all such invisible shuffles is the Galois group: it is the measured ambiguity in the roots.

Solving by radicals means peeling that ambiguity away one clean layer at a time. Adjoining an $n$ th root resolves a cyclic (and so abelian) amount of ambiguity. The question "can I peel it all away?" becomes "can the group be torn down into abelian sheets?" For the quintic, the symmetry group $S_{5}$ contains the rigid, indivisible block $A_{5}$ — a knot of symmetry with no normal cuts. You can never unwind it abelian-layer-by-layer, so no radical formula exists.

Worked Examples

Galois group of $x^2 - 2$ over $\mathbb{Q}$

Given:

f (x) = x^{2} - 2

, base field

K = Q

. Step 1. Roots are

\pm 2

, so the splitting field is

L = Q (2)

, of degree

[L : K] = 2

. Step 2. An automorphism

σ \in Gal (L / K)

must send

2

to a root of

x^{2} - 2

, i.e.

σ (2) \in {2, - 2}

. Step 3. Two choices: identity and the conjugation

2 \mapsto - 2

. Both preserve field structure. Result:

Gal (Q (2) / Q) ≅ Z_{2}, ∣ G ∣ = 2

This group is abelian, hence solvable — consistent with $x^{2} - 2$ being solvable by radicals ( $x = \pm 2$ ).

Galois group of $x^3 - 2$ over $\mathbb{Q}$

Given:

f (x) = x^{3} - 2

, base field

K = Q

. Step 1. Roots:

32, ω 32, ω^{2} 32

, where

ω = e^{2 π i /3}

. Splitting field

L = Q (32, ω)

. Step 2. Degree:

[Q (32) : Q] = 3

and

[Q (32, ω) : Q (32)] = 2

, so

[L : Q] = 6

. Step 3. Thus

∣ G ∣ = 6

. Automorphisms permute the three roots and act faithfully, so

G

embeds in

S_{3}

; since

∣ G ∣ = 6 = ∣ S_{3} ∣

, we get

G ≅ S_{3}

. Step 4. Check solvability via [5]:

S_{3} ▹ A_{3} ▹ {e}

with quotients

Z_{2}

and

Z_{3}

, both abelian. Result:

Gal (L / Q) ≅ S_{3}, ∣ G ∣ = 6, solvable

So $x^{3} - 2$ is solvable by radicals (Cardano's cubic formula applies).

Common Misconceptions

"Galois proved the quintic has no solutions." False. Every quintic has five complex roots (Fundamental Theorem of Algebra). What Galois proved is that no general formula in radicals exists. Specific quintics like

x^{5} - 1 = 0

x^{5} - 2 = 0

are solvable by radicals — their Galois groups are solvable. The unsolvability is about the general equation with symbolic coefficients.

A solvable quintic

Consider $f (x) = x^{5} - 2$ . Its splitting field is $Q (52, ζ_{5})$ where $ζ_{5} = e^{2 π i /5}$ . The Galois group has order 20 and is isomorphic to the Frobenius group $F_{20} = Z_{5} ⋊ Z_{4}$ , which has the subnormal series ${e} ◃ Z_{5} ◃ F_{20}$ with abelian quotients $Z_{5}$ and $Z_{4}$ . Hence $F_{20}$ is solvable, and indeed $x = 52 ζ_{5}^{k}$ are radical expressions. This single example refutes the belief that all degree-five equations resist radicals.

"The Galois group is just the permutations of roots, $S_{n}$ ." Only for the generic polynomial. For a specific polynomial, hidden algebraic relations among roots shrink the group to a proper subgroup of

S_{n}

. For instance

x^{4} + 1

over

Q

has Galois group

Z_{2} \times Z_{2}

, not

S_{4}

, because the four roots satisfy extra rational relations.

Connections

Galois's work is the keystone of modern algebra and reaches far beyond polynomials. It generalizes the impossibility results of Ruffini–Abel Theorem, which showed that the quintic is unsolvable; Galois explained why and exactly when. The machinery rests on Group Theory (he essentially invented normal subgroups and the word "group" in its modern sense) and Field Theory. The Fundamental Theorem of Galois Theory mirrors duality structures seen in Covering Space Theory (fundamental group ↔ covers) and in Galois Connections in order theory. It directly resolves the classical Compass and Straightedge Construction problems (trisecting the angle, doubling the cube) by translating constructibility into degree-power-of-two field extensions.

Real-World Applications

Error-correcting codes. Reed–Solomon codes (used in CDs, DVDs, QR codes, and deep-space communication by NASA) are built over finite fields $F_{q}$ , whose extension structure is Galois-theoretic.
Cryptography. Elliptic-curve cryptography and AES's MixColumns step operate in finite fields $F_{2^{8}}$ ; their Galois (Frobenius) automorphisms underpin the arithmetic.
Coding theory hardware. Galois Field arithmetic units appear in RAID storage and Galois/Counter Mode (GCM) authenticated encryption used in TLS.
Constructibility verdicts. Galois theory settles the impossibility of trisecting a general angle and constructing a regular 7-gon, and certifies that a regular 17-gon is constructible (Gauss).
Inverse Galois problem in computer algebra. Software like GAP and Magma compute Galois groups to factor polynomials and test solvability.
Differential equations. Differential Galois theory (Picard–Vessiot) decides whether ODEs have solutions in "closed form," extending Galois's solvability idea to integrals and the Risch algorithm in symbolic integration.
Number theory and the Langlands program. Galois representations link arithmetic to analysis and were central to Wiles's proof of Fermat's Last Theorem.

Historical Development

1770 — Lagrange studies resolvents and permutations of roots, planting the idea that symmetry governs solvability, but cannot resolve the quintic.
1799–1813 — Paolo Ruffini publishes incomplete proofs that the general quintic is unsolvable by radicals; largely ignored.
1824 — Niels Henrik Abel gives the first rigorous proof of quintic unsolvability (Ruffini–Abel Theorem), independent of Galois.
1829 — At age 17, Galois submits his first memoir on solvability to the Académie; Cauchy, the referee, loses or mislays it.
1830 — Galois publishes short notes in Férussac's Bulletin; his father commits suicide that year; he is admitted to the École Normale.
1831 — Galois submits a revised memoir; Poisson reviews it and declares it "incomprehensible," recommending fuller exposition. Galois is imprisoned for political activity.
30 May 1832 — Galois is mortally wounded in a duel; dies the next day, aged 20, after a night spent writing out his theory and a famous testamentary letter to Auguste Chevalier.
1843–1846 — Joseph Liouville studies the manuscripts, verifies their correctness, and publishes them in his Journal, finally bringing Galois's ideas to the mathematical world.
1870 — Camille Jordan's Traité des substitutions systematizes Galois theory into the foundation of modern group theory; later Dedekind, Weber, and Artin recast it in the field-automorphism language used today.

Subtleties & Limits

The Inverse Galois Problem remains open: Is every finite group the Galois group of some Galois extension of $Q$ ? It is known to hold for all solvable groups (Shafarevich's theorem), all symmetric and alternating groups, and 25 of the 26 sporadic simple groups, but the general case is unresolved. Related deep questions include the structure of the absolute Galois group $Gal (\overline{Q} / Q)$ — a monstrous profinite object whose understanding is a central goal of the Langlands program and Grothendieck's theory of dessins d'enfants. Galois theory in positive characteristic and for inseparable extensions also requires careful generalization (Artin–Schreier theory, purely inseparable extensions).