**Niels Abel** was a Norwegian mathematician who proved the impossibility of solving algebraically the general equation of the fifth degree.

- In this difficult time Abel was growing up in Gjerstad in south-east Norway.
- What had been a good school was in a bad state when Abel arrived.
- When a new mathematics teacher Bernt Holmboë joined the school in 1817 things changed markedly for Abel.
- Abel began to study university level mathematics texts and, within a year of Holmboë's arrival, Abel was reading the works of Euler, Newton, Lalande and d'Alembert.
- Holmboë was convinced that Abel had great talent and encouraged him greatly taking him on to study the works of Lagrange and Laplace.
- Holmboë was able to help Abel gain a scholarship to remain at school and Abel was able to enter the University of Christiania in 1821, ten years after the university was founded.
- Holmboë had raised money from his colleagues to enable Abel to study at the university and he graduated in 1822.
- While in his final year at school, however, Abel had begun working on the solution of quintic equations by radicals.
- Degen asked Abel to give a numerical example of his method and, while trying to provide an example, Abel discovered the mistake in his paper.
- At the University of Christiania Abel found a supporter in the professor of astronomy Christopher Hansteen, who provided both financial support and encouragement.
- In 1823 Abel published papers on functional equations and integrals in a new scientific journal started up by Hansteen.
- In Abel's third paper, Solutions of some problems by means of definite integrals he gave the first solution of an integral equation.
- Abel was given a small grant to visit Degen and other mathematicians in Copenhagen.
- Returning to Christiania, Abel tried to get the University of Christiania to give him a larger grant to enable him to visit the top mathematicians in Germany and France.
- Abel began working again on quintic equations and, in 1824, he proved the impossibility of solving the general equation of the fifth degree in radicals.
- By this time Abel seems to have known something of Ruffini's work for he had studied Cauchy's work of 1815 while he was an undergraduate and in this paper there is a reference to Ruffini's work.
- Abel sent this pamphlet to several mathematicians including Gauss, whom he intended to visit in Göttingen while on his travels.
- In August 1825 Abel was given a scholarship from the Norwegian government to allow him to travel abroad and, after taking a month to settle his affairs, he set out for the Continent with four friends, first visiting mathematicians in Norway and Denmark.
- On reaching Copenhagen, Abel found that Degen had died and he changed his mind about taking Hansteen's advice to go directly to Paris, preferring not to travel alone and stay with his friends who were going to Berlin.
- In Copenhagen Abel was given a letter of introduction to Crelle by one of the mathematicians there.
- Abel met Crelle in Berlin and the two became firm friends.
- This proved the most useful part of Abel's whole trip, particularly as Crelle was about to begin publishing a journal devoted to mathematical research.
- Abel was encouraged by Crelle to write a clearer version of his work on the insolubility of the quintic and this resulted in Recherches sur les fonctions elliptiques Ⓣ(Researches on elliptic functions) which was published in 1827 in the first volume of Crelle's Journal, along with six other papers by Abel.
- While in Berlin, Abel learnt that the position of professor of mathematics at the University of Christiania, the only university in Norway, had been given to Holmboë.
- With no prospects of a university post in Norway, Abel began to worry about his future.
- Crelle's Journal continued to be a source for Abel's papers and Abel began to work to establish mathematical analysis on a rigorous basis.
- It had been Abel's intention to travel with Crelle to Paris and to visit Gauss in Göttingen on the way.
- However, news got back to Abel that Gauss was not pleased to receive his work on the insolubility of the quintic, so Abel decided that he would be better not to go to Göttingen.
- It is uncertain why Gauss took this attitude towards Abel's work since he certainly never read it - the paper was found unopened after Gauss's death.
- the first possibility is that Gauss had proved the result himself and was willing to let Abel take the credit.
- Crelle was detained in Berlin and could not travel with Abel to Paris.
- Abel therefore did not go directly to Paris, but chose to travel again with his Norwegian friends to northern Italy before crossing the Alps to France.
- In Paris Abel was disappointed to find there was little interest in his work.
- Abel's theorem states that any such sum can be expressed as a fixed number p of these integrals, with integration arguments that are algebraic functions of the original arguments.
- Abel's theorem is a vast generalisation of Euler's relation for elliptic integrals.
- In Berlin, Abel borrowed some money and continued working on elliptic functions.
- Crelle tried to persuade Abel to remain in Berlin until he could find an academic post for him and he even offered Abel the editorship of Crelle's Journal.
- However, Abel wanted to get home and by this time he was heavily in debt.
- Abel was appointed to this post which improved his position a little.
- In 1828 Abel was shown a paper by Jacobi on transformations of elliptic integrals.
- Abel quickly showed that Jacobi's results were consequences of his own and added a note to this effect to the second part of his major work on elliptic functions.
- Abel continued to pour out high quality mathematics as his health continued to deteriorate.
- Abel travelled by sled to visit his fiancée again in Froland for Christmas 1828.
- Crelle was told and he redoubled his efforts to obtain an appointment for Abel in Berlin.
- He succeeded and wrote to Abel on the 8 April 1829 to tell him the good news.
- It was too late, Abel had already died.
- After Abel's death his Paris memoir was found by Cauchy in 1830 after much searching.
- Also after Abel's death unpublished work on the algebraic solution of equations was found.
- In this same year 1830 the Paris Academy awarded Abel and Jacobi the Grand Prix for their outstanding work.

Born 5 August 1802, Frindöe (near Stavanger), Norway. Died 6 April 1829, Froland, Norway.

View full biography at MacTutor

Prize Abel, Algebra, Group Theory, Origin Norway

Definitions: 1

Lemmas: 2

Propositions: 3

**O’Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive