Mathematics 1824

Memoir on Algebraic Equations, Proving the Impossibility of Solving the General Equation of the Fifth Degree

Niels Henrik Abel

The general quintic has no formula in radicals.

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In depth · the introduction

For 250 years, mathematicians hunted for a formula to solve any fifth-degree equation. A 21-year-old Norwegian proved the formula they were chasing could never exist.

The big idea

You probably learned the quadratic formula: feed it the numbers in an equation like x² + bx + c = 0, and out come the answers, built from those numbers using add, subtract, multiply, divide, and square roots. There are matching (messier) formulas for cubic equations (x³…) and quartic ones (x⁴…), found in the 1500s.

The natural next step was a formula for the quintic — the fifth-degree equation. Generation after generation tried, and failed. Abel showed why: there is no such formula, and there can't be. For equations of degree five and higher, the roots simply cannot, in general, be written down using only those familiar operations and roots. He didn't fail to find the formula — he proved nobody ever could.

How it came about

Niels Henrik Abel grew up poor in Norway, his talent spotted by a teacher who paid for his studies. As a teenager he thought he had SOLVED the quintic — then caught his own mistake, and the failure pointed him at a deeper question: maybe the reason no one could find the formula was that none existed.

An Italian, Paolo Ruffini, had argued the same thing from 1799, but his proofs had a hole he never quite filled. Abel filled it. In 1824 he printed his proof as a six-page pamphlet at his own expense — so cramped, to save money, that few could follow it. Two years later he rewrote it in full for a new German mathematics journal. He died of tuberculosis at 26, just as recognition was finally arriving.

Why it mattered

This was one of the first times mathematics proved that something is IMPOSSIBLE — not "we haven't found it yet," but "it cannot be found, ever." That is a different and harder kind of knowledge. To prove it, Abel (and soon Galois) invented ways of thinking about the symmetry of an equation's solutions, which grew into group theory — today one of the central languages of all mathematics, physics, and chemistry.

A way to picture it

Think of a maze where the only moves you're allowed are the "radical" ones: add, subtract, multiply, divide, and take roots. For degree-2, 3, and 4 equations there's always a path from the coefficients to the answer using just those moves. Abel showed that for the general degree-5 equation, the answer sits in a room with no door reachable by those moves at all. The answer exists — every such equation does have five roots — but no sequence of the allowed steps will ever reach it.

Where it sits

Solving equations is one of the oldest threads in mathematics, running from Babylonian tablets through the Renaissance Italians who cracked the cubic and quartic. Abel ended the quest for the quintic formula; Évariste Galois — another tragic young genius, dead at 20 in a duel — explained exactly which equations are solvable and which aren't, founding Galois theory. Their symmetry-based thinking became group theory, which later let physicists describe the symmetries of nature and underpins modern cryptography.

The original document

Original source text

The pamphlet & its claim

N. H. Abel · Mémoire sur les équations algébriques · Christiania (Grøndahl), 1824 · 6 pp.

Full French title

Mémoire sur les équations algébriques, où l'on démontre l'impossibilité de la résolution de l'équation générale du cinquième degré.

[Memoir on algebraic equations, in which the impossibility of solving the general equation of the fifth degree is demonstrated.]

Abel paid for the printing himself and held the work to six pages, which made it almost impenetrably condensed. Two years later he reworked the argument at length for the inaugural volume of Crelle's Journal.

The theorem (1826 statement)

Beweis der Unmöglichkeit, algebraische Gleichungen von höheren Graden als dem vierten allgemein aufzulösen · Crelle's Journal 1 (1826): 65–84

It is impossible to solve, in general, algebraic equations of degree higher than the fourth by radicals — that is, by a finite formula built from the coefficients using only addition, subtraction, multiplication, division and the extraction of roots.

The result applies to the GENERAL equation, where the coefficients are treated as independent symbols. Particular quintics with special structure (for example x⁵ − 2 = 0) can still be solved by radicals; what Abel ruled out is a single formula that works for every quintic.

The shape of the proof

Abel supposes, for contradiction, that a solution by radicals exists, and asks what form it must take. He shows that every radical appearing in such a solution can be written as a rational function of the roots of the equation and the coefficients.

He then studies how those expressions behave when the roots are permuted among themselves — counting how many distinct values a function of the roots can take. A theorem of Cauchy limits this number; Abel shows the required radical structure cannot be reconciled with it for degree five.

[ … ]

The two demands collide: the assumed radical formula forces a function of the five roots to take a number of values that Cauchy's count forbids. The contradiction shows no such formula can exist.

N. H. Abel · Christiania · 1824

The gap Abel closed

Paolo Ruffini had argued for the same impossibility from 1799 onward, but his proofs assumed without justification that any radical in a solution must already be a rational function of the roots. Abel's contribution was to PROVE this step — now often called Abel's theorem on the form of radicals — turning a plausible assumption into a rigorous one.

Abel knew of the difficulty of the general problem but, by his own account, did not know Ruffini's work in detail when he wrote the 1824 pamphlet.