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The quadratic formula solves every equation of degree 2. Similar formulas exist for degrees 3 and 4. For degree 5, mathematicians searched for 250 years and found nothing, and Abel proved that no general formula exists. A French teenager, Evariste Galois, then explained which equations can be solved by a formula and why. This post tells his story from reliable sources and explains his central idea with small examples you can work through.
Who was Evariste Galois?
The MacTutor biography records that Galois was born in Bourg-la-Reine, near Paris, on 25 October 1811, and died in Paris on 31 May 1832. His mother taught him at home until he was 12. At the Lycee Louis-le-Grand he took his first mathematics class in February 1827 and was quickly absorbed. His teachers' reports called him "singular", "bizarre" and "original".
He failed the entrance examination of the Ecole Polytechnique twice, in 1828 and 1829. In July 1829, a few weeks before the second attempt, his father, the mayor of Bourg-la-Reine, took his own life after a political scandal. Galois entered the Ecole Normale instead, but was expelled after attacking its director in a newspaper letter.
Why was his work rejected?
His first paper, on continued fractions, appeared in April 1829, when he was 17. He sent papers on solving equations to the Academie des Sciences, with Cauchy as referee. In February 1830 he submitted a new version for the Grand Prize; it went to Fourier, the Secretary, who died soon after, and it was never found. A third version, sent in January 1831, was rejected by Poisson, who wrote that the argument was "neither sufficiently clear nor sufficiently developed to allow us to judge its rigour", while encouraging him to write a fuller account.
He was arrested twice in 1831 for republican protests and held in Sainte-Pelagie prison. On 30 May 1832 he was wounded in a duel, and he died the next day. MacTutor notes that the story of him writing all his mathematics on the last night is exaggerated.
What was Galois' big idea?
Instead of hunting for a formula, Galois asked how the roots of an equation can be shuffled. Take all the rational relations between the roots, such as "their sum is 5". The permutations of the roots that keep every such relation true form what we now call the group of the equation. In modern terms, the equation can be solved by radicals exactly when this group can be broken down in a special step-by-step way; such groups are called solvable.
Worked example: shuffling roots
x² − 5x + 6 = 0. The roots are 2 and 3. The relation "first root minus 2 equals 0" is rational and true for (2, 3) but false if you swap them. So the only allowed permutation is "leave them alone". The group has 1 element, which matches the fact that the roots are already rational.
x² − 2 = 0. The roots are √2 and −√2. Their sum is 0 and their product is −2; both stay true when you swap them. No rational relation tells them apart, so the group has 2 elements: do nothing, or swap.
Counting shuffles. With n roots there are at most n! permutations: 2, 6, 24 and 120 for degrees 2, 3, 4 and 5. Our post on permutations and counting explains n!.
Degree 5. For a general quintic, all 120 permutations are allowed. That group is not solvable, so there is no general formula, which agrees with Abel's theorem.
Common mistakes:
Thinking "no formula" means "no roots". Every quintic has five roots in the complex numbers; you just cannot always write them with +, −, ×, ÷ and nth roots.
Thinking no quintic can be solved by radicals. x⁵ − 32 = 0 has the root 2. Only the general case fails.
Allowing shuffles that break a relation. Every rational relation must survive, not just the sum and product.
To revise how roots relate to coefficients, see our guide to quadratic equations and their roots.
How did his work survive?
His brother and his friend Auguste Chevalier copied his papers and sent them to Gauss and Jacobi. MacTutor records that in September 1843 Joseph Liouville told the Academie he had found in them a solution "as correct as it is deep", and published them in his journal in 1846. Peter Neumann's The Mathematical Writings of Evariste Galois (2011) gives the papers in French and English.
The story of the man who proved the general quintic has no formula is in our post on Niels Henrik Abel.
What can a student take from Galois' way of working?
Change the question when the old one is stuck. He studied symmetry instead of formulas.
Write so others can follow. Poisson rejected his work for lack of clarity, not because it was wrong.
Rejection is not the final verdict. His ideas were recognised eleven years after his death.
When a definition like "group" will not sink in, the Fermi Concept Library has every concept in mathematics with its own lesson and its place in the map.
List all six orderings of three objects a, b, c, and find which ones keep the relation "a + b = 2c" true when a = 1, b = 3, c = 2.
