1. Introduction: The Young Genius Abel

In the history of mathematics, there are a few geniuses who passed away at a young age but left a decisive impact on future generations. Among them, the Norwegian-born Niels Henrik Abel stands alongside Évariste Galois as one of the most famous tragic geniuses. In his short life of only 26 years, he proved that “there is no general algebraic solution for equations of degree five or higher,” a problem that had plagued mathematicians for centuries.

In this article, we will delve into Abel’s life, driven by his passion for mathematics despite poverty and illness, and his monumental achievements such as “Abelian groups” and “Abelian integrals.”

2. Abel’s Life: Poverty and the Blooming of Talent

2.1 Early Childhood and Meeting His Mentor Holmboe

Niels Henrik Abel was born on August 5, 1802, in the small Norwegian village of Finnøy, as the son of a pastor. Norway at the time was economically impoverished, and Abel’s family was no exception.

His destiny changed significantly when he entered the Cathedral School in Oslo in 1817 and met his mathematics teacher, Bernt Michael Holmboe. Holmboe immediately recognized Abel’s extraordinary talent and taught him university-level advanced mathematics. By devouring the works of masters like Euler, Lagrange, and Laplace, Abel quickly absorbed cutting-edge mathematics.

  flowchart TD
    A["1802: Born in Norway"] --> B["1817: Meets Holmboe"]
    B --> C["1821: Enters Royal Frederick University"]
    C --> D["1824: Self-publishes the impossibility of the quintic equation"]
    D --> E["1825-1827: European study tour (Berlin, Paris)"]
    E --> F["1829: Dies of tuberculosis at age 26"]

2.2 The Death of His Father and a Heavy Burden

In 1820, when Abel was 18, his father passed away. His father had deepened political conflicts and died leaving behind heavy debts. As a result, Abel had to bear the heavy responsibility of supporting his mother and six siblings. Despite extreme poverty, with the support of his mentor Holmboe and friends, he entered the Royal Frederick University (now the University of Oslo) in 1821.

3. The Impossibility of the Quintic Equation Formula

Abel’s first great achievement was related to algebraic equations. Quadratic equations have had a known solution formula since ancient times, and formulas for cubic and quartic equations were discovered in 16th-century Italy. However, no one could find a general formula for the quintic equation for nearly 300 years after that.

Initially, Abel thought he had discovered the formula for the quintic equation and wrote a paper. However, during the peer-review process by Holmboe and others, he realized his mistake. Subsequently, he arrived at the opposite idea: “Could it be that there is no general formula using only the four basic arithmetic operations and radicals for equations of degree five and above?”

Finally, in 1824, he completely proved this fact. Today, this is known as the Abel-Ruffini theorem (since the Italian mathematician Paolo Ruffini had previously published an incomplete proof).

It is generally impossible to express the roots of this equation in a finite number of terms using only $a, b, c, d, e, f$, the four arithmetic operations, and radicals.

4. Journey to Europe and Meeting Crelle

In 1825, Abel obtained a scholarship from the Norwegian government and had the opportunity to study in continental Europe. His goal was to visit Paris, the center of mathematics at the time, and Göttingen, where the great mathematician Carl Friedrich Gauss resided.

Abel sent his paper to Gauss, but Gauss ignored it without even reading it. Giving up on meeting Gauss, Abel headed to Berlin.

In Berlin, he met August Leopold Crelle, a civil engineer and passionate mathematics enthusiast. Impressed by Abel’s talent, Crelle launched the world’s first specialized mathematics journal, Journal für die reine und angewandte Mathematik (commonly known as Crelle’s Journal). Abel contributed multiple papers to its inaugural issue, making his name known in the European mathematical community.

5. Setback in Paris and Cauchy’s Neglect

In 1826, Abel arrived in Paris. Here he submitted a paper to the French Academy of Sciences on a “Broad theorem concerning transcendental functions,” which could be considered his masterpiece. This paper contained groundbreaking content that would later be known as Abel’s theorem.

However, misfortune struck again. The great mathematician Augustin-Louis Cauchy, who was assigned to review it, misplaced Abel’s paper in a pile of documents in his room and never reviewed it.

Driven out by despair, lack of funds, and the creeping disease of tuberculosis, Abel was forced to leave Paris.

6. Return Home and Tragic End

In 1827, extreme poverty awaited Abel upon his return to Norway. Unable to find a permanent academic position, he wrote papers frantically in the little time he had left.

On April 6, 1829, watched over by his fiancée and friends, Abel passed away at the young age of 26.

Tragically, just two days after his death, a letter arrived from Crelle. It stated that a mathematics professorship at the University of Berlin had been secured for Abel. By the time his talent received its rightful recognition, it was already too late.

7. Abel’s Mathematical Legacy

The achievements Abel left behind have taken root in all areas of modern mathematics.

7.1 Abelian Group

In abstract algebra, one of the most fundamental and important structures is the “Group”. Among groups, those in which the result does not change even if the order of operations is swapped are called Abelian groups.

By definition, a group $G$ is called an Abelian group if the following commutative law holds for any elements $a, b$ in $G$:

  flowchart LR
    A["Group"] --> B{"Does the commutative law hold?"}
    B -- "Yes: a*b = b*a" --> C["Abelian Group"]
    B -- "No: a*b ≠ b*a" --> D["Non-Abelian Group"]
    
    C --> E["Example: Integers with addition"]
    D --> F["Example: Matrix multiplication"]

7.2 Abelian Integrals and Abelian Functions

The subject of Abel’s Paris memoir, the Abelian integral, is a generalization of integrals involving algebraic functions. After his death, this theory was developed by Jacobi and others, growing into magnificent theories such as Abelian varieties in algebraic geometry.

7.3 Abel’s Limit Theorem

Abel also left a significant mark in the field of analysis. It is a theorem regarding the convergence of infinite series.

This theorem is still frequently used today in theories such as analytic continuation and asymptotic expansion.

8. Conclusion

Niels Henrik Abel’s life was truly fitting of the word “tragedy.” However, the passion he poured into mathematics and the numerous theorems he produced will never fade. The theories he left behind continue to provide new inspiration to mathematicians to this day.