The quadratic formula has been taught in schools for so long that most people forget how unusual it is. Given any quadratic equation , it returns the two roots:
Two thousand years before this formula was written down, mathematicians knew how to solve quadratics geometrically. By the 1500s, Italian mathematicians (Cardano, Tartaglia, Ferrari) had extended the formula to cubics and quartics. The cubic and quartic formulas exist; they’re horrible to look at, but they exist. Each one expresses the roots in terms of the coefficients using only the basic operations and nth roots.
For the next three centuries, mathematicians searched for a similar formula for degree-5 polynomials — the quintic. Prizes were offered. Famous mathematicians tried and failed. By 1800 it was widely suspected that no such formula existed, but no one could prove it.
Then, in 1832, a 20-year-old named Évariste Galois settled the question forever the night before he was killed in a duel. His proof was so deep, so unprecedented, that the mathematical establishment couldn’t read it for thirty years. When they finally did, the field of Galois theory was born — a theory that doesn’t just answer the quintic question but reorganizes most of algebra around the concept of symmetry.
This article tells the story of what Galois did, why it mattered, and how a teenager’s last-night-on-earth letter became one of the most important documents in the history of mathematics.
The historical setup
Solving polynomials by formula is an old problem. The Babylonians could solve quadratics around 2000 BCE, geometrically. The Greeks had clean geometric methods. By the 9th century, al-Khwarizmi had a complete algebraic treatment of quadratics — and the word “algebra” comes from the Arabic title of his book.
Cubics resisted for longer. In 1535, in a famous public mathematical contest in Italy, Niccolò Tartaglia revealed his solution to a particular family of cubic equations. Cardano coaxed the secret out of him under a promise of confidentiality and then published it in his 1545 book Ars Magna — sparking one of the great priority disputes in mathematical history. Cardano’s formula for the cubic, even cleaned up, is genuinely intimidating.
His student Ferrari extended the work to the quartic (degree 4) within a few years. The quartic formula is even worse to write out, but exists.
For the quintic, nothing worked. Lagrange in the 1770s analyzed the problem deeply and concluded that the methods used for cubics and quartics fundamentally fail at degree 5. Ruffini, in 1799, attempted a proof of impossibility; his argument had gaps. Niels Henrik Abel, in 1824, gave a rigorous proof that no general quintic formula in radicals exists. This is now called the Abel-Ruffini theorem.
But Abel’s proof, while correct, didn’t explain why. It showed that quintics in general aren’t solvable; it didn’t characterize which specific quintics might be. That’s what Galois did.
Galois’s life
Évariste Galois was born in 1811 outside Paris. By age 14 he was reading advanced mathematical works for fun and producing original results in number theory. By 16 he had submitted a paper to the French Academy of Sciences on the solvability of equations.
His timing was unlucky. The paper was lost. He resubmitted; that paper was rejected by Cauchy as too obscure. He submitted again to a Grand Prize competition; the manuscript was lost when the judge died before reading it. A fourth submission was rejected by Poisson with a note saying the work was “incomprehensible.”
Galois was, by his own account and his contemporaries’, a difficult young man — politically radical, prone to confrontation, distrustful of authority. He failed the entrance exam for the École Polytechnique twice. He was expelled from the École Normale for political agitation. He was imprisoned in 1831 for an alleged threat against the king.
In May 1832 he was challenged to a duel — the cause has never been definitively established; theories range from a romantic entanglement to political setup. Knowing he might die, he spent the night before writing a long mathematical letter to his friend Auguste Chevalier. The letter sketches what we now call Galois theory and ends with the famous postscript: “Plus tard, on déchiffrera tout ce gâchis” — “Later, all this mess will be deciphered.”
He was shot on the morning of 30 May 1832 and died the next day. He was 20.
What the theory says
Galois’s central insight was to attach a group to every polynomial — a precise algebraic object now called the Galois group. The group encodes the symmetries of the polynomial’s roots: the ways the roots can be permuted while preserving the algebraic relations among them.
The Galois group of a polynomial determines its solvability. Specifically:
Theorem (Galois, 1832). A polynomial is solvable by radicals if and only if its Galois group is solvable.
A “solvable group” is one that can be built up from cyclic groups by a sequence of normal extensions — a precise group-theoretic condition. Solvable groups include all abelian groups and many non-abelian ones, but they don’t include certain particularly complicated groups.
For a generic polynomial of degree , the Galois group is the symmetric group of all permutations of the roots. We can ask: is solvable?
- is solvable (it’s just ).
- is solvable (it has a normal subgroup of order 3 with cyclic quotient).
- is solvable, just barely — by an elegant chain involving the Klein four-group.
- is not solvable. The alternating group inside it is simple — meaning it has no proper normal subgroups — and large enough to break the inductive construction.
Galois proved this. So the generic quintic has Galois group , which is not solvable, so the generic quintic is not solvable in radicals. The same argument applies to every degree .
The argument doesn’t say “we couldn’t find a formula.” It says: no formula can possibly exist, for fundamental algebraic reasons. The structure of the symmetric group for forbids it.
A specific quintic that is solvable
Galois’s theorem is sharper than just an impossibility result. It tells you exactly which quintics are solvable.
The polynomial is solvable. Its roots are the five fifth roots of 2: , where is a primitive 5th root of unity. Its Galois group is a group of order 20 — solvable.
The polynomial is not solvable. You can compute its roots numerically (one is approximately 1.1673), but you cannot write them as a finite expression in symbols. Its Galois group is .
This is the kind of distinction Galois theory makes. It’s not just “degree 5 is bad” — some degree-5 polynomials are perfectly solvable, others are fundamentally unsolvable, and the Galois group tells you which is which.
Other consequences
Galois theory does much more than answer the solvability question. The framework — extending fields by adjoining roots, studying the resulting symmetries — turned out to be one of the most flexible tools in algebra.
Constructibility with compass and straightedge. Greek geometry left three classical problems unsolved for millennia: trisecting an arbitrary angle, doubling the cube, and squaring the circle. Galois theory proves all three are impossible. The numbers constructible with compass and straightedge form a very specific field, and the impossibility of trisection comes from the Galois group of the relevant polynomial having the wrong structure.
Construction of regular polygons. Gauss had shown in 1796 that the regular 17-gon is constructible — a famous result. Galois theory gives a complete characterization: the regular -gon is constructible if and only if is a power of 2 times distinct Fermat primes. So the 3-, 4-, 5-, 6-, 8-, 10-, 12-, 15-, 16-, 17-, 20-, … gons are constructible; the 7-, 9-, 11-, 13-, 14-gons are not.
Algebraic numbers. A number is algebraic if it satisfies some polynomial with integer coefficients. Galois theory provides the framework for studying such numbers — splitting them into towers of fields, classifying their conjugates, computing class numbers and ideal class groups. Algebraic number theory, as a modern subject, is downstream of Galois.
Inverse Galois problem. Given a finite group , can you find a polynomial whose Galois group is ? This is one of the major open problems in algebra. The answer is known to be yes for many specific groups but is unresolved in general.
Why the theory is beautiful
Galois theory is consistently cited as one of the most beautiful subjects in mathematics. The reason is the depth of its central insight: that algebraic problems have group-theoretic shadows, and that the shadows are often easier to study than the originals.
You start with a polynomial — a concrete computational object. You attach a group — an abstract structural object. The group’s properties (solvability, simplicity, transitivity) translate directly into the polynomial’s properties (solvability in radicals, irreducibility over various fields, geometric constructibility). One layer of mathematics encodes the other, in a way that turns hard analytic questions into tractable algebraic ones.
This kind of move — find an algebraic invariant that encodes an analytic question — became one of the most productive techniques in 20th-century mathematics. Algebraic topology, algebraic geometry, K-theory, motivic cohomology — all are descendants of the Galois move. You don’t ask “can this thing be solved?” You ask “what is the algebraic invariant that encodes whether it can be solved?” Then you study the invariant.
What Galois left
Galois died with three published papers totaling about 50 pages. His mathematical reputation in 1832 was negligible. None of the papers had been understood by his readers; his major work was unpublished and floating among friends.
The reason we know his work today is the letter to Chevalier, written the night before the duel. Chevalier preserved it and showed it around. Joseph Liouville, fifteen years later in 1846, finally edited and published Galois’s manuscripts. Even then, the work was hard to follow — group theory as a formal subject didn’t exist yet, and Galois’s notation was idiosyncratic. It took decades for mathematicians to fully digest what he had done.
By the 1860s, Cayley had formalized group theory. Camille Jordan’s Traité des substitutions (1870) was the first textbook on Galois theory. By 1900 the framework was standard graduate material. By 2000 it was undergraduate material. Today every algebra student learns Galois theory, often in their first year of mathematical studies.
The lesson, perhaps, is about how mathematical priority works. Galois had results that were a generation ahead of what anyone else had. He couldn’t communicate them to his contemporaries, partly because of his own difficult personality and partly because the conceptual framework needed to read him hadn’t been built. He died young, in a senseless duel, with his work unrecognized.
What survived was 50 pages of writing, mostly written in a single night by a 20-year-old who knew he might be killed in the morning. From those pages, an entire branch of mathematics grew. Whether that’s a triumph or a tragedy depends on what you think mathematics is for.
But the pages are still there, and the theorems are still true, and the polynomial remains stubbornly unsolvable for the same reason Galois noticed: is not solvable, and never will be.
Frequently asked
Is there really no formula for the quintic?
There is no formula in radicals — that is, no formula using only the four basic operations and nth roots. There are formulas using more general functions (the Bring radical, elliptic modular functions), but these aren't elementary. So 'no formula' specifically means 'no formula like the quadratic formula.'
Did Galois really die in a duel at 20?
Yes. Évariste Galois died on 31 May 1832 from wounds sustained in a duel the day before. He was 20. The night before, he wrote out his mathematical results in a long letter to his friend Auguste Chevalier — much of modern Galois theory comes from those notes. The reason for the duel remains debated: probably political, possibly romantic, certainly avoidable.
Why are groups the right tool here?
Because the symmetries of a polynomial's roots — the ways you can permute them while preserving algebraic relationships — form a group. If this group has a particular structure (it's solvable), the polynomial is solvable in radicals. If the group is more complicated (e.g. the alternating group A_5), no formula in radicals exists. Galois was the first to make this connection.