For over two centuries, mathematicians had trouble in finding a solution to the quintic equation, that is until Niels Henrik Abel formulated a theory. Abel was a Norwegian mathematician born on August 5, 1802, and his talent and potential in the field of mathematics was already present at a young age, leading him to become a pioneer in the field as he developed solutions to problems nobody could answer.
In 1813, Niels Abel entered the Cathedral school in Oslo, Norway, where his brilliance in mathematics was recognized by his teacher, Bernt Michael Holmboe in 1817. Holmboe encouraged Abel to study advance levels of math and even went to the measure of tutoring this prodigy after school. He also introduced Niels to the classics of mathematical literature and presented him with original problems to solve. In preparation for his own research, Niels Abel studied the works of several notable mathematicians during his time, including Sir Isaac Newton and Leonhard Euler. After the death of his father in 1820, Abel and his family suffered from an emotional and financial crisis, fortunately enough, Holmboe helped Abel obtain a scholarship and, along with some of his friends, contributed and raised enough funds for Niels to enroll in the University of Oslo in 1821.
After receiving his preliminary degree in 1822, Abel furthered his studies and eventually published his first papers on integrals and functional equations, in which he, for the first time in history, formulated and solved an integral equation. In midst of waiting for the Norwegian government to grant him a fellowship in 1824, “he published, at his own expense, his proof of the impossibility of solving algebraically the general equation of the fifth degree, which he hoped would bring him recognition” (Britannica), this became known as Abel’s Impossibility Theorem that states that “in general, polynomial equations higher than fourth degree are incapable of algebraic solution in terms of a finite number of additions, subtractions, multiplications, divisions, and root extractions” (MathWorld), his impossibility theory describes what is known today as quintic equations. Unfortunately, when he sent his pamphlet to one of his contemporaries, Carl Friedrich Gauss, it was immediately dismissed as Gauss failed to see that Abel already settled the famous two centuries old question.
Niels Abel met August Leopold Crelle in the winter of 1824 to 1825 in Berlin. Crelle was a self-taught enthusiast of mathematics and Abel’s mentor, he founded the Journal für die reine und angewandte Mathematik, which translates to “Journal for Pure and Applied Mathematics”, or famously known today as Crelle’s Journal. The first volume was published in 1826 and consisted of Abel’s papers of a more elaborate explanation on his work on the quintic formula. Soon, Abel’s theory of elliptic functions, complex functions which generalized usual trigonometric formulas, was also demonstrated in later volumes.
While in the world center of mathematics of the time, Paris, Abel developed and finished a vital paper on the theory of integrals of algebraic functions along with other mathematicians. Abel’s theorem was the main outcome of the paper and became the foundation “for the later theory of Abelian integrals and Abelian functions, a generalization of elliptic function theory to functions of several variables” (Britannica).
On returning home to Oslo, Niels Abel suffered from the disease of debt and tuberculosis, he ended up teaching in 1828 to earn some money. Although he was in a state of poverty and illness, Abel continued to write more papers and develop more theories and solutions, including the theory of polynomial equations with Abelian groups and the theory of elliptic functions. This helped him and his works to become recognized by more mathematical centers of his time and secure him a profession suitable for his talents. By April 6, 1829, Niels Henrik Abel became fatally ill and passed away at the age of 26, leaving behind notable works that successfully impacted and advanced the field of mathematics.
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