Niels Henrik Abel

Introduction

In the Royal Park of Oslo in Norway, there stands a monument in memory of the great mathematician Niels Henrik Abel, who has made significant contribution to the theory of elliptic functions and the solutions of higher degree equations.

At Teenage

Abel was born in a poor family. He did not go to school until he was 13 years old. At the age of 15, Abel met his first mathematics teacher, Bernt Holmboe, who was only 7 years elder than him, but who already had a strong background in mathematics. He taught Abel all he knew in mathematics, and Abel completed the study of elementary mathematics in a year or so. Then Abel moved on to study advanced mathematics.

In his final year in college, Abel began to work on a famous mathematical problem, namely, the solution of a general quintic equation. This problem, which had remained to be open for more than 200 years, was generally considered to be very difficult. He submitted his conjecture for the problem to Holmboe, but Holmboe did not understand his conjecture completely. He simply advised Abel to concentrate on his study and prepare for the university entrance examination.

At University

With the help of Holmboe, Abel successfully entered the University of Christiania with a scholarship. Abel restarted his work on the famous problem in his first year at university. He once believed that he had solved it completely, and submitted a paper to the mathematics professor Rasmussen and the astronomy professor Christopher Hansteen.

Both professors could not say much about the paper, so they sent the paper to the Danish mathematician Ferdinand Degen. Degen found some mistakes in the paper, but nevertheless gave Abel some important advice in his reply letter: “Mr. Abel, although you have not solved the problem yet, you showed excellent talents in mathematics. I hope you will work not only on the solution of quintic equations, but also on other parts of mathematics, such as the theory of elliptic functions, which is going to have great influence on the development of mathematics.”

In fact, Abel was not depressed; rather, he was touched by Degen’s sincerity and took his advice. Abel carried on his effort in his university study and later finished a treatise on integration with a remarkable success.

At Foreign Countries

In 1824, Abel, at the age of 22, eventually conquered the problem of solving quintic equations. His conclusion, which was very pioneering at his days, was that “it is not possible to solve a general quintic equation by only addition, subtraction, multiplication, division and radicals on its coefficients”. At that time, most mathematicians could hardly accept his proof. On one hand, he was too young and had failed to solve the same problem just a few years ago. On the other hand, in order to minimize the cost of publication, his paper was written very briefly with a lot of intermediate steps omitted, making it very difficult for other mathematicians to understand.

Failing to be recognized by the majority, Abel went to Germany, hoping that mathematicians there would understand his proof, and that he could earn a living there. He intended to visit Gauss, the leading mathematician of the time, but eventually gave up the idea, realising that Gauss did not even send a reply after he sent Gauss his paper.

Later, Abel went to France, where many top mathematicians lived. However, he was disappointed to find that the French mathematicians did not quite like young scholars, and that the leading position of France in mathematical research was already beginning to decline.

Meanwhile, Abel was in deep trouble. He had run out of scholarship and unfortunately caught tuberculosis owing to cold weather. Despite the cruel reality, Abel continued his research and came up with many new ideas in mathematics. He decided to return to Christania to write down all his new discoveries.

Returning Home

Back to Christania, Abel worked on his new ideas day and night. Following the publication of many innovative papers, Abel finally caught the attention of many mathematicians in Europe. When they knew that Abel only managed to make a hard living by a temporary job of tutoring schoolchildren, they wrote a letter to the King of Norway, requesting him to improve Abel’s living conditions and provide him with better working conditions.

But it was too late. Abel’s health continued to deteriorate. He died before any good news came.

His short life

On 6th April 1829, Abel passed away at the age of 26. Nevertheless, he left his successors two invaluable assets, namely, the theory of algebraic equations and the theory of elliptic functions, which greatly influenced the subsequent development of mathematics.

On 28th June 1830, the Paris Academy awarded Abel the Grand Prix for his outstanding works. Abel’s achievements finally won worldwide recognition.