On August 5th, 1802, Soren George Abel and Marie Simonsen gave birth to their son Niels Henrik Abel in Nedstrand, Norway (Encyclopedia Britannica). Niels grew up in a very religious family, with his father Soren having a degree in theology and philosophy and serving as a pastor at Finney. Additionally, Niels’ grandfather Hans Mathias Abel was a pastor at Gjerstad, where Soren was a chaplain during his childhood (Wolfram Research). Following Hans’ passing, Soren took over as pastor in Gjerstad.
Niels’ mother, Marie, grew up in Risor. Her father, Niels Henrik Saxlid Simonsen, was considered the wealthiest man in Risor. In Gjerstad, Marie was accustomed to a luxurious life and was responsible for organizing parties and gatherings. According to Encyclopedia Britannica, Marie was not very involved in raising their children and instead spent a significant amount of time with alcohol. Soren, who was elected as a representative of Storting, established a connection with the Cathedral School in Christiania (also known as Oslo). Instead of sending his eldest son Hans, Soren decided to send Niels there (as stated in Encyclopedia Britannica).
In 1815, at the age of 13, Niels entered the Cathedral School. The following year, his older brother Hans joined him at Olso. The two brothers not only shared a room but also attended classes together. Despite Hans boasting about receiving better grades, there was a differing opinion about which of the Able brothers was the smarter one. This was evident in 1818 when Olso employed Brent Michael Holmboe, a Mathematics professor with a fresh viewpoint on the brothers.
Brent noticed Niels’ skill in mathematics and believed that he had a promising future in the subject. Impressed by Niels’ abilities, Brent decided to provide him with private math lessons (Stubhaug). Meanwhile, while Niels was away at school, his family, the Abels, experienced a difficult period. In 1818, Niels’ father Soren found himself involved in a heated theological debate with Stener Johannes Stenersen.
This theological argument, documented by the Encyclopedia Britannica, revolved around Stener’s catechism and religious perspectives on the New Testament. It gained significant media attention during that period. However, in the midst of this controversy, Soren unintentionally offended the wrong individuals, jeopardizing his own situation. Shortly after this dispute with Stener, Soren turned to excessive drinking. Sadly, after two years of unending alcohol consumption, Soren succumbed to his addiction, leaving his wife and children in a profound state of sadness.
The family members each responded differently to Soren’s death. The sons, Hans and Niels, took opposite paths. Hans became deeply depressed and felt the need to go back home to mourn, while Niels received a scholarship from Brent Holmboe, his esteemed professor (according to Encyclopedia Britannica). Brent’s support went beyond securing a scholarship for Niels to stay in Olso; he also raised funds and found ways to financially assist Niels in attending the Royal Frederick University (as stated by Wolfram Research). Consequently, Niels enrolled at the Royal Frederick University in 1821.
At just 19 years old, Niels had already established himself as one of the top mathematicians in the field, earning the admiration of many. During his first year at Royal Frederick University, Niels embarked on his first project, exploring the quintic equation in radicals. In 1822, he graduated with outstanding credentials in mathematics, though his performance in other subjects was average. Prior to graduating, Niels believed he had discovered the solution to a longstanding problem that had persisted for 250 years (Stubhaug). His confidence in his work on the quintic equation in radicals was supported by renowned mathematicians such as Ferdinand Degen and Christopher Hansteen, who found no flaws or inaccuracies in Niels’ findings.
Despite being overwhelmed by the idea that a 250-year-old problem was finally solved, Degen requested Niels to demonstrate another example where this solution would work (Wolfram Research). However, when Niels reviewed his work with the new example, he discovered that his solution was incorrect, leaving the 250-year-old problem still unsolved. Following his graduation from Royal Frederick University, Niels Abel faced difficulties in finding a new residence, forcing his professors to provide financial assistance until he could secure housing.
Despite his financial instability, he remained committed to his mathematical ambitions and resolutely continued making significant progress in the field. Niel’s breakthroughs began to emerge in various aspects of his life after 1822. In 1823, Niel achieved a significant milestone by publishing an article in “Magazin for Naturvidenskaberne,” which was Norway’s inaugural scientific journal. Encouraged by this accomplishment, he went on to publish numerous subsequent articles in the same publication. However, upon realizing that his articles needed to be accessible to the general public, Niel decided to write an entire paper in French.
According to the Encyclopedia Britannica, the paper was reported to present a broad representation of the possibility of integrating differential formulas. Following his numerous publications, Niels was offered an opportunity to go to Copenhagen and work on Fermat’s Last Theorem. While in Copenhagen, Niels visited his uncle Peder Tuxen and it was during this visit that he met his future spouse Christine Kemp. In the midst of these endeavors, Niels also published one of his initial remarkable works titled “Memoir on Algebraic Equations”. Eventually, in 1823, Niels successfully proved his quintic equations’ radical nature, leading to the subsequent naming of this achievement as the “Abel-Ruffini Theorem”.
Abel’s first notable work, Memoire Sur Les equations algebriques ou on demonstrating impossibility de la resolution de equation general du cinquieme degre, was published in 1824 while he was studying these two languages. In this memoir, Abel proves the impossibility of solving the general equation of the fifth degree (Wikipedia). Additionally, because Abel could no longer receive money from his professors, he began receiving scholarship funds. Despite this, Abel remained anxious about the publication of his extensive memoir on integration, for which he had initially sought university support.
The manuscript containing the idea for the great Abelian theorem, which later disappeared, circulated with his fellowship documents in a public archive. Although the formula he indicated is not fully understandable, it undoubtedly contains the concept for the result. His momentum, as described by Ore, is enduring and lasts longer than bronze. In September 1825, he obtained permission to travel abroad and went on a journey to Berlin and the Alps with four companions to study geology. He planned to visit Carl Gauss by first going to Gottingen and then continuing to Paris.
While on his trip, Abel spent a total of four months in Berlin and developed a close friendship with August Crelle. During this period, August Crelle was preparing to publish his mathematical journal called Journal fur die Reine und Angewandte Mathematik. Within the first year of its publication, Abel made significant contributions to seven articles in this journal. After leaving Berlin, Abel continued his journey to Leipzig, Freiberg, Dresden, Prague, Vienna, Trieste, Venice Verona Bolzano Innsbruck Luzern and Basel. It was during his time in Freiberg that Abel achieved notable progress and conducted research on functions such as elliptic and hyperelliptic functions. He also explored a newly discovered type now known as abelian functions.
In July of 1826, Abel traveled to his final destination, Paris. He sent a portion of his work to Berlin to be published in Crelle’s journal while reserving his most significant work for the French Academy of Sciences. This particular work would later become known as the great Abelian theorem. Unfortunately, his work was not well known in France and eventually faded into obscurity. Due to financial problems, Abel’s time on his study abroad trip was shortened, causing him to return to Berlin in January 1826. By May 1827, he had returned to Norway.
The reason his scholarship was not renewed was because they believed his travels had been unsuccessful. In need of money, he turned to tutoring and continued working with Crelle’s journal. When Niels Abel returned to Norway, he had contracted tuberculosis during his time in Paris. In 1828, on Christmas, he traveled to see his fiancee, but his illness worsened during the journey. Despite this, the couple managed to enjoy the holidays before Abel eventually passed away on April 6, 1829. Following his death, a letter from Crelle arrived announcing that Abel had been offered a professorship at the University of Berlin.
Niels Henrik Abel’s death marked the conclusion of an exceptional and promising journey in mathematics. The French Academy published his memoir in 1841, recognizing Abel’s remarkable achievements and acknowledging his future potential. Thanks to several scholarships, he had not only delved into different languages like German and French but also explored foreign lands to expand his mathematical knowledge. As a result, Niels Henrik Abel emerged as one of the most influential mathematicians from Scandinavia.
Abel’s career as a mathematician was short-lived due to his premature death at the age of 27. However, he made significant contributions to the field. At just 16 years old, Abel provided a proof of the Binomial Theorem for all numbers, not only rational numbers. Previously, this theorem had only been proven for rational numbers by Leonhard Euler.
At 19 years old, Abel invented a branch of mathematics with numerous applications in physics. He developed a theory that demonstrated the absence of an algebraic solution for finding the roots of quintic numbers or any higher degree equation. This solution was highly sought after and its discovery at such a young age by Abel was truly impressive.
After proving that solutions do not exist for degrees four or higher, Abel gained recognition in the mathematics community and went on to produce Abel’s Theorem of Convergence.
This statement by Abel is known as one of his most well-known works. According to this statement, if a real power series converges for a positive argument value, then the domain of uniform convergence extends up to and including this point. Additionally, the continuity of the sum of the function also extends up to and including this point (Wolfram Alpha).
At 21 years old, Abel had already published several papers on functional equations and integrals, including the first solution to an integral equation. Additionally, one year later, he demonstrated the impossibility of algebraically solving equations of the fifth degree. After achieving this breakthrough, Abel fell seriously ill and went back to Norway, where he continued his research on equation theory and elliptic functions.
His work on elliptic functions had significant implications for the entire theory of elliptic functions. One crucial aspect of his discoveries was the examination of the inverses of these functions. Abel recognized that in order to find the solution, he needed to study the inverse of the elliptic function. Niels Henrik Abel was at the forefront of a mathematical transformation. In the early nineteenth century, mathematical analysis primarily focused on formulas and evolved into a more concept-centered approach. The changes involved various aspects such as subject matter, notions of rigor, methods, and objects (Sorensen page 40).
The content was divided into distinct disciplines and sub-disciplines. Algebra encompassed the theory of equations, while the remaining portion of analysis concentrated on studying real or complex functions. Abel’s elliptic function fell within this particular category. With the introduction of new complex mathematical techniques, there was a transition toward a more arithmetic-focused foundation for analysis. As a result, functions were no longer described by their formal relationships but by numerical values.
Abel significantly contributed to the transformation of analysis by redefining objects by class. He hoped to gain recognition for his work by writing a paper and sharing it with various mathematicians. However, he was unaware that Paolo Ruffini, an Italian mathematician, had already published the same proof years ago. Despite this, Abel’s original draft contained elements that did not align with a valid proof. Ferdinand Degen, a mathematician, suggested that Abel focus on working in the field of elliptic integrals.
Without this suggestion, Abel may not have made his discoveries. He traveled the world in an attempt to gain recognition for his work but was unsuccessful. Upon returning home, he discovered that he had contracted tuberculosis but continued to seek recognition for his efforts. Eventually, mathematicians began to take notice of Abel’s work and recognize its significance in the field. Abel was an early proponent of providing proofs for various equations and even gave the proof for the binomial theorem at the age of 16.
This theorem was significant as it built upon Euler’s concept, which was only applicable to rational numbers. Abel’s theorem expanded upon Euler’s theorem by stating that the binomial theorem is applicable to all numbers. Following the binomial theorem, Abel embarked on his mathematical journey at the age of 19 by attempting to solve the quintic equation. The quintic equation takes the form of [x^5 + bx^4 + cx^3 + dx^2 + ex + f = 0]. Abel demonstrated that there exists no general solution for finding the algebraic roots of a quintic equation when the degree exceeds four. To accomplish this, Abel introduced a new branch of mathematics known as group theory.
This area of math became crucial for both math and physics. Group theory, which is the study of groups, is a method used to analyze abstract and physical systems. It reveals the significance of symmetry for physics. Abel commenced his research on elliptic functions and his theory of equations. These concepts require precise technical formulas, without which these equations cannot be accurately described. The discovery of calculus enabled the computation of integrals involving square roots as well as expressions involving degrees of three and four.
The calculation of ellipse arc lengths, known as elliptic integrals, posed a problem until Abel proposed a new approach. Instead of investigating the integrals directly, he transformed them into inverse functions, which provided a fresh perspective. This led to the development of the theory of elliptic functions, which generalized trigonometry and established properties akin to those of sine, cosine, and tangent. Abel also delved into equation theory, studying quadratic equations that were taught to Babylonians and Greeks by the Arabs.
Around 1500 A.D., Scipione del Ferro discovered the solution to the cubic equation. Ferro intentionally kept this information secret due to its immense value. Over time, the equation was passed on to various mathematicians. One such mathematician was Gerolamo Cardano, who published a work called Ars Magna. This publication not only included the cubic equation but also provided a solution for equations of the fourth degree. As soon as the solution for the fourth-degree equation became known, mathematicians began their quest to find solutions for fifth and higher degrees. Although Abel tried to prove the existence of solutions for higher degrees, he eventually demonstrated that a solution did not exist.
