# Mathematician:Niels Henrik Abel

## Contents

## Mathematician

Norwegian mathematician who died tragically young.

Made significant contributions towards algebra, analysis and group theory.

Best known for proving the impossibility of solving the general quintic in radicals (Abel-Ruffini Theorem).

Due to a series of administrative mishaps and personal blunders by various influential mathematicians, he was not recognised for what he was until too late.

An important influence on the founding of *Crelle's Journal*, the first volume for which he contributed several papers.

He was finally appointed to a chair of mathematics in Berlin, but by that time he had died of tuberculosis at the age of $26$.

## Nationality

Norwegian

## History

- Born: 5 Aug 1802, Frindöe (near Stavanger), Norway
- 1815: Sent to the Cathedral School in Christiania (now Oslo) with his older brother
- 1817: Became a pupil of Bernt Holmboe
- 1820: Death of Abel's father
- 1821: Enter University of Christiania with assistance from Bernt Holmboe
- 1822: Graduated
- 1826: Went to Paris
- Died: 6 April 1829, Froland, Norway

## Theorems and Definitions

### Definitions

- Abelian Function
- Abelian Group
- Abelian Integral
- Abel's Integral Equation
- Abelian Mean
- Abel's Mechanical Problem
- Abel's Series
- Abel Summation Method
- Abel Summability
- Abelian Theorem

Definitions of concepts named for **Niels Henrik Abel** can be found here.

### Results

- Abel's Generalisation of Binomial Theorem
- Abel's Lemma, also known as Abel's Transformation and Abel's Partial Summation Formula
- Abel's Limit Theorem
- Abel's Test
- Abel's Test for Uniform Convergence
- Abel's Theorem
- Abel-Ruffini Theorem (with Paolo Ruffini)

Results named for **Niels Henrik Abel** can be found here.

## Publications

- 1823 (?):
*Solutions of some problems by means of definite integrals*

- 1824:
*Mémoire sur les équations algébriques ou 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 proven)

- 1826:
*Beweis eines Ausdruckes, von welchem die Binomial-Formel ein einzelner Fall ist.*(*J. reine angew. Math.***Vol. 1**: pp. 159 – 160)

- 1826:
*Mémoire sur une Propriété Générale d'une Classe Très-Étendue de Fonctions Transcendantes*(lost by Augustin Louis Cauchy for years and not published till $1841$)

- 1827:
*Recherches sur les fonctions elliptiques*

## Notable Quotes

*It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils.*

*My eyes have been opened in the most surprising manner. If you disregard the very simplest cases, there is in all of mathematics not a single infinite series whose sum has been rigorously determined. In other words, the most important parts of mathematics stand without a foundation. It is true that most of it is valid, but that is very surprising. I struggle to find a reason for it, an exceedingly interesting problem.*

## Critical View

*Abel has left mathematicians enough to keep them busy for $500$ years.*

*All of Abel's works carry the imprint of an ingenuity and force of thought which is amazing. One may say that he was able to penetrate all obstacles down to the very foundations of the problem, with a force of thought which appeared irresistible ... He distinguished himself equally by the purity and nobility of his character and by a rare modesty which made his person cherished to the same unusual degree as was his genius.*

*Abel, the lucky fellow! He has done something everlasting! His ideas will always have a fertilizing influence on our science.*

## Sources

- John J. O'Connor and Edmund F. Robertson: "Niels Henrik Abel": MacTutor History of Mathematics archive

- 1937: Eric Temple Bell:
*Men of Mathematics*: Chapter $\text{XVII}$ - 1965: J.A. Green:
*Sets and Groups*... (previous) ... (next): $\S 4.4$. Gruppoids, semigroups and groups - 1966: Richard A. Dean:
*Elements of Abstract Algebra*... (previous) ... (next): $\S 1.3$ - 1970: B. Hartley and T.O. Hawkes:
*Rings, Modules and Linear Algebra*... (previous) ... (next): $\S 1.1$: The definition of a ring: Definitions $1.1 \ \text{(b)}$ - 1971: Allan Clark:
*Elements of Abstract Algebra*... (previous) ... (next): Introduction - 1978: John S. Rose:
*A Course on Group Theory*... (previous) ... (next): $0$: Some Conventions and some Basic Facts - 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next): Entry:**Abel, Niels Henrik**(1802-29) - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {A}.27$: Abel ($\text {1802}$ – $\text {1829}$) - 1997: Donald E. Knuth:
*The Art of Computer Programming: Volume 1: Fundamental Algorithms*(3rd ed.) ... (previous) ... (next): $\S 1.2.6$: Binomial Coefficients - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (next): Entry:**Abel, Neils Henrik**(1802-29) - 2004: Ian Stewart:
*Galois Theory*(3rd ed.) ... (previous) ... (next): Historical Introduction: Polynomial Equations - 2008: David Joyner:
*Adventures in Group Theory*(2nd ed.) ... (previous) ... (next): Where to begin... - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**Abel, Neils Henrik**(1802-29) - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**Abel, Neils Henrik**(1802-29)