Niels Henrik Abel was born in Norway in 1802. In 1815, Abel entered the Cathedral School, aged 13. Hans Mathias Abel, joined him the next year. They shared apartment and had classes together. His brother did, in general, get better grades than Niels Henrik Abel. A new mathematics teacher, Bernt Michael Holmboe, was appointed in 1818. He gave the students mathematical tasks to do at home. Seeing Abel's talent in mathematics he encouraged him to study the subject to an advanced level. He also gave Abel private lectures after school. In 1826 his first notable work was published in the first volume of Crelle’s Journal, though it had been previously published on a smaller scale two years earlier. In this journal, Abel’s work of a proof proving that it was impossible to solve the quintic equation using radicals was displayed. Abel was able to visit Germany and France in 1825. He stayed in Berlin for six months where he met and became friends with August Leopold Crelle. Abel’s work contributed greatly to the success of the journal that Crelle published.
Later, in Freiberg, Abel worked on his researches in the theory of elliptic and hyperelliptic functions. He also studied “Abelians” at that time. When Abel moved to France in 1826, he was able to meet the country’s greatest mathematicians. But, because his work was not well known, these mathematicians did not take very much note of him.
However, Abel lived a short life, which cut short his mathematical genius. Abel died in 1829 of Tuberculosis. But, during his lifetime, he helped new areas of mathematics to become ‘open’ and ushered in very advanced study of functions.
While in Paris, Abel had contracted tuberculosis. For Christmas 1828, he traveled by sled to again visit his fiancée in Froland. He became seriously ill on the journey, although a temporary improvement allowed the couple to enjoy the holiday together. Crelle, at the same time, had been searching for a new job for Abel in Berlin, and did manage to have him appointed professor at a university. Crelle wrote to Abel on April 8, 1829 to tell him the good news, but Abel had died two days earlier.
At the age of 16, Abel gave a proof of the binomial theorem valid for all numbers, extending Euler's result which had only held for rationals. At age 19, he showed there is no general algebraic solution for the roots of a quintic equation, or any general polynomial equation of degree greater than four, in terms of explicit algebraic operations. To do this, he invented (independently of Galois) an extremely important branch of mathematics known as group theory, which is invaluable not only in many areas of mathematics, but for much of physics as well. Among his other accomplishments, Abel wrote a monumental work on elliptic functions which, however, was not discovered until after his death. When asked how he developed his mathematical abilities so rapidly, he replied "by studying the masters, not their pupils."
The early death of this talented mathematician, of whom Adrien-Marie Legendre said "quelle tête celle du jeune Norvégien!" ("what a head the young Norwegian has"), cut short a career of extraordinary brilliance and promise. Under Abel's guidance, the prevailing obscurities of analysis began to be cleared, new fields were entered upon and the study of functions so advanced as to provide mathematicians with numerous ramifications along which progress could be made. His works, the greater part of which originally appeared in Crelle's Journal, were edited by Holmboe and published in 1839 by the Norwegian government, and a more complete edition by Ludwig Sylow and Sophus Lie was published in 1881. The adjective "abelian", derived from his name, has become so commonplace in mathematical writing that it is conventionally spelled with a lower-case initial "a" (e.g., abelian group, abelian category, and abelian variety). On April 6, 1929, four Norwegian stamps were issued for the centenary of Abel's death. His portrait appears on the 500-kroner banknote (version V) issued during 1978–1985. On June 5, 2002, four Norwegian stamps were issued in honour of Abel two months before the bicentenary of his birth. There is also a 20-kroner coin issued by Norway in his honour. A statue of Abel stands in Oslo, and crater Abel on the Moon was named after him. In 2002, the Abel Prize was established in his memory.
"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"