Niels Henrik Abel

Niels Henrik Abel was a pioneering Norwegian mathematician who transformed the field of algebra.

Born in 1802 in Nedstrand, Norway, he was the son of a pastor named Søren Georg Abel. His early life was spent in the rectory at Gjerstad, where he received his primary education from his father. While his brother Hans Mathias initially attended the Cathedral School in Christiania, Niels eventually followed him there in 1815. It was at this school that his mathematical potential was first recognized. A teacher named Bernt Michael Holmboe provided him with private lessons and later secured a scholarship for his university studies. This support allowed Abel to enter the Royal Frederick University in 1821, where he quickly became the most knowledgeable mathematician in Norway.

Abel's most significant contribution involved the quintic equation, a type of algebraic equation of the fifth degree. For over 250 years, mathematicians had attempted to find a general solution for these equations using radicals, which are expressions involving roots. In 1821, Abel initially believed he had found such a solution. His work was reviewed by professors Søren Rasmussen and Christopher Hansteen, and later by Carl Ferdinand Degen in Copenhagen. However, while attempting to provide a numerical example for Degen, Abel discovered an error in his own formulas. This realization led to his groundbreaking 1823 discovery: it is impossible to solve the general quintic equation using radicals. This result is now formally known as the Abel–Ruffini theorem.

Beyond his work on equations, Abel made major advancements in the study of functions. He was an innovator in the field of elliptic functions, which are complex mathematical functions used in various branches of science. He also discovered a specific class of mathematical objects now called Abelian functions. Most of his profound discoveries were made during a very brief period of only six or seven years of active work. His research was often published in Crelle's Journal, a mathematical publication in Berlin. Abel contributed seven articles to the journal during its first year, helping to ensure its success. His ability to navigate complex analytical landscapes was described by Degen as finding "Magellanian thoroughfares" through a vast analytical ocean.

Abel's career was marked by both brilliant success and extreme financial hardship.

Although he was highly talented, he lived in poverty for much of his working life. He traveled through several European cities, including Copenhagen, Berlin, and Paris, to connect with other scholars. In Copenhagen, he met Christine Kemp, and the two became engaged to be married.

However, his travels were often cut short by a lack of funds. He once applied for a scholarship to study in Germany and France, but he was instead granted funds to stay in Christiania to learn German and French. By 1827, his financial situation had become so difficult that he had to return to Norway and take out a loan from Norges Bank.

Despite these struggles, Abel's intellectual impact was immense. The French mathematician Charles Hermite remarked that Abel had left enough mathematical material to keep scholars busy for five hundred years. Another mathematician, Adrien-Marie Legendre, expressed awe at the young Norwegian's intellect. Even when his work was submitted to the French Academy of Sciences, it was sometimes overlooked due to his modesty or the difficulty of the subject matter. One of his major theorems regarding the addition of algebraic differentials was submitted to the academy for review by Augustin-Louis Cauchy, but it remained largely unknown during his lifetime.

Abel's life was cut short by illness. He died in 1829 at the age of 26 from tuberculosis. Because he died so young, many of his potential future discoveries remained unwritten. However, the work he did complete changed the direction of mathematics forever. His proofs provided definitive answers to problems that had remained open since the 16th century. He moved the study of algebra away from simply searching for solutions and toward understanding the fundamental properties of equations and functions. His legacy is preserved in the many mathematical terms and theorems that bear his name.

Today, the study of Abel's work connects to many modern fields of mathematics and physics. His discoveries in elliptic and Abelian functions are essential to higher-level analysis and geometry. The Abel–Ruffini theorem remains a foundational concept in Galois theory and abstract algebra. By proving what could not be done, Abel opened new doors for mathematicians to explore different methods of reasoning. His life serves as a powerful example of how intense intellectual curiosity can reshape human knowledge, even when faced with significant personal adversity.