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Your textbook has a formula for quadratics. There are longer ones for cubics and quartics. So why does the list stop at degree 4? Niels Henrik Abel, a young Norwegian with almost no money, proved that it has to. This post tells his story from reliable sources, works through the cubic formula on one example so you can see what "solving by radicals" means, and explains what Abel's theorem does and does not say.
Who was Niels Henrik Abel?
The MacTutor biography records that Abel was born near Stavanger, Norway, on 5 August 1802, and died at Froland on 6 April 1829. He grew up during years of war, blockade and hunger in Norway. At the Cathedral School in Christiania (now Oslo) he was an ordinary pupil until a new teacher, Bernt Holmboe, arrived in 1817. Within a year Abel was reading Euler, Newton and d'Alembert.
His father died in 1820, leaving the family in debt. Holmboe raised money from colleagues so that Abel could enter the University of Christiania in 1821.
How did a mistake lead to his theorem?
In his last year at school Abel believed he had found a formula for the general quintic. MacTutor records that he sent it to the Danish mathematician Ferdinand Degen, who asked for a numerical example. While trying to produce one, Abel found the error.
By 1824 he had proved the opposite: the general equation of the fifth degree cannot be solved by radicals. He printed the proof at his own expense as a pamphlet, cut to six pages to save money. Its opening line notes that others had tried to prove the impossibility, "but, if I am not mistaken, they have not succeeded up to the present". A fuller German version appeared in 1826 in the first volume of Crelle's Journal. Paolo Ruffini had published an impossibility proof in 1799 which, according to MacTutor's Ruffini biography, was remarkable but had one gap.
What does "solving by radicals" mean?
It means writing the roots using only the coefficients, +, −, ×, ÷ and square, cube or higher roots. MacTutor's Cardan biography records that the methods for the cubic and quartic were published in Cardano's Ars Magna in 1545. Here is the cubic method on one equation.
Worked example: a cubic by radicals
Solve x³ + 6x − 20 = 0.
Split the unknown. Put x = u + v. Then x³ = u³ + v³ + 3uv(u + v), so x³ − 3uv·x − (u³ + v³) = 0.
Match coefficients. Comparing with x³ + 6x − 20 = 0 gives −3uv = 6, so uv = −2, and u³ + v³ = 20.
Make a quadratic. u³ and v³ have sum 20 and product (uv)³ = −8, so they are the roots of t² − 20t − 8 = 0. Then t = 10 ± √108 = 10 ± 6√3.
Take cube roots. (1 + √3)³ = 1 + 3√3 + 9 + 3√3 = 10 + 6√3, so u = 1 + √3. Likewise v = 1 − √3.
Add. x = (1 + √3) + (1 − √3) = 2.
Check. 2³ + 6(2) − 20 = 8 + 12 − 20 = 0, and uv = (1 + √3)(1 − √3) = −2.
Abel's theorem says no recipe of this kind, however long, works for every equation of degree five.
Common mistakes:
Getting the sign of uv wrong in step 2; it comes from −3uv = 6.
Thinking no quintic is solvable. x⁵ − 1 = 0 has the root 1. Abel's result is about the general equation, and Galois later explained which ones can be solved, as our post on Evariste Galois shows.
Thinking no roots exist. They exist and can be found numerically; there is just no general radical formula.
For revision on how roots relate to coefficients, see our guide to quadratic equations and their roots.
How did his life end?
MacTutor records that Abel sent his pamphlet to Gauss, who never read it; it was found unopened after Gauss's death. In Berlin he met August Crelle, who became a firm friend and published his work. In Paris in 1826 his long memoir on what is now called Abel's theorem was given to Cauchy and Legendre to referee and then seemed to have been lost. Back in Norway, poor and in debt, he fell ill on a sled journey to his fiancee at Froland at Christmas 1828. The Abel Prize biography says consumption, the old name for tuberculosis, took him at 26. Crelle wrote on 8 April 1829 to say he had secured him a post in Berlin; Abel had died two days earlier.
The Abel Prize, established by the Norwegian Government in 2002, now honours mathematicians in his name each year.
What can a student take from Abel's way of working?
Test your own work. A single example exposed his first error.
Prove the limits of a method. Showing something cannot be done is a real result.
Find your people. Holmboe and Crelle made his work possible.
If the algebra of roots and coefficients still feels shaky, the Fermi Concept Library has every concept in mathematics with its own lesson, so you can strengthen one idea at a time.
Use the same steps to solve x³ + 3x − 4 = 0 by radicals, then check your answer by substitution.
