Quintic Equations, French Politics, and Abstract Algebra

The familiar quadratic formula produces the two solutions of any equation of the form ax²+bx+c. And much more elaborate formulas exist for the three solutions of cubic equations (those with an x³ term) and the four solutions of quartic equations (with an x⁴ term). Similarly, equations with an x⁵ term—quintic equations—always have five solutions. But over the centuries, a formula for those solutions failed to emerge.

In 1821, Norwegian mathematician Niels Abel (1802-1829) thought he had found it. He in fact sent his formula to the Danish Academy of Sciences before discovering a crucial flaw. This made him change course to a proof of impossibility, building off the work of Joseph-Louis Lagrange (1736-1813). Lagrange, investigating how permutations (rearrangements) of the roots of polynomials affect certain expressions made with those roots, had observed properties common to quadrics, cubics, and quartics, but lacking in quintics. Abel’s proof relates these symmetries to what is now considered a particular field extension. Complex numbers are a simple example of this; they use the four basic operations to “extend” the set of real numbers by i, the square root of -1. The field in Abel’s proof extends the set of rational numbers by the five roots of a cubic.

After publishing his proof in 1824, Abel toured Europe, attempting to publicize it. But his work was terse to the point of incomprehensibility, and a previous, incorrect proof by Paolo Ruffini (1765-1822) had made many famous mathematicians skeptical. The French Academy of Sciences misplaced the copy sent to them. Abel would spend the last few years of his life trying unsuccessfully to find work as a professor, though he did do more major research on fields. 

By the time Abel died of tuberculosis in 1829, a young Parisian student named Évariste Galois (1811-1832) was investigating the solutions of all polynomials. Deeply inspired by Lagrange, he was trying to formalize the connection between roots of a polynomial and the set of polynomials made with the field extension by those roots. His approach added the nonrational numbers of the extension to the rational ones in a series of steps; the structure of this “tower” determines whether the roots are writable with radicals and basic operations. This method shows that there exist polynomials with such unwritable roots for all degrees five and greater; thus no simple formula exists for any of them.

On the advice of his enthusiastic mentor Louis-Paul-Émile Richard, Galois submitted a mémoire, or paper, to the Academy of Sciences. But his mathematics, though brilliant, was habitually idiosyncratic and confusing—for instance, he used the term “group” to refer to three slightly different concepts relevant to the tower of extensions. It is unsurprising that the Academy largely ignored it. Galois, however, was angry at the lack of recognition. And then his application to the elite École Polytechnique was rejected for similar reasons, further infuriating him.

Galois turned instead to the École Normale. It was a teachers’ college, not much to his liking, but it gave scholarships and his family needed the money. He eventually, barely, got in. Not long afterwards, he submitted the same mémoire for a prize put out by the Academy of Sciences; once again nothing came of it. Around this time, he began becoming politically involved. 

In July of 1830, the Bourbon king Charles X tried to sign a series of authoritarian decrees. In response, Paris exploded into three days of street fighting. Many students participated, though not Galois (the director of the École Normale had imposed a strict curfew and the walls were too high to climb). The pace of events surprised Charles X, who soon abdicated. Though the new king, his cousin Louis-Philippe, was moderate and likable, the more extreme revolutionaries felt betrayed. They hadn’t wanted any king at all. Many of them would soon join the Société des Amis du Peuple, which eventually did set off the brief, unsuccessful June Rebellion dramatized in Hugo’s Les Miserables.

Galois was one of these, though he did not live to see the June Rebellion. Also around this time, he published an indignant letter in the school newspaper, got expelled, gave unintelligible mathematics tutoring, and submitted his work yet again to the Academy of Sciences. As the contemporary mathematician Sophie Germain put it, he “kept up his capacity for being rude.” Then, in May of 1831, he was arrested for a threatening toast to the king at a republican banquet. 

Though acquitted, he did not change his behavior in the slightest. On July 14 he took prominent part, heavily armed, in a Bastille Day protest. This time he was sentenced to nine months in the Sainte-Pélagie prison. Increasingly depressed and resentful, he nonetheless continued refining his research. He also wrote a bitterly sarcastic preface for his mémoire. It strongly criticizes the mathematical establishment as opposed to true innovation, blaming them for, among other things, Abel’s death. Galois proudly admits the difficulty of his research, calling it “analysis of analysis…the highest calculations.”

Galois was paroled in March. Two months later, on May 30, 1832, he was shot in a duel, dying the next day. The surrounding circumstances remain controversial. Contemporary sources variously identify his opponent, though it does seem clear that he shared Galois’s politics. Much of the evidence supports the most common account—that the quarrel was over a woman. In any case, Galois’s letters in the days leading up to the duel show that he expected to die. 

The most famous of these hurried letters was addressed to Auguste Chevalier, a close friend from the École Normale. In it, Galois summed up all his major work with solutions to polynomials. This letter is one of the best statements of Galois’s ideas, incorporating a number of revisions, but it still has logical leaps and ragged edges. The final line is “Subsequently there will be, I hope, some people who will find it to their profit to decipher all this mess.”

Chevalier managed to get his friend’s mémoire published in 1846. In the following decades, mathematicians such as Arthur Cayley (1821-1895) and Camille Jordan (1838-1922) built on and clarified Galois’s ideas—for instance, precisely and abstractly defining groups—to create the new domain of group theory. This perspective lies at the root of abstract algebra, and can be combined with analytic geometry to prove that not all angles are trisectable. In addition, group theory provides a rigorous characterization of symmetry with applications in areas such as crystallography. It took another generation, but Galois’s “mess” indeed proved quite profitable. 

Image Credit:

http://euclid.nmu.edu/~joshthom/Teaching/MA412/

https://commons.wikimedia.org/wiki/File:Niels_Henrik_Abel.jpg

https://commons.wikimedia.org/wiki/File:Evariste_galois.jpg

https://commons.wikimedia.org/wiki/File:E._Galois_Letter.jpg

Sources:

Rigatelli, Laura Toti, and John Denton. Evariste Galois 1811-1832. Basel: Birkhauser, 1996.

Smith, David Eugene. A source book in Mathematics, by David Eugene Smith. New York: McGraw-Hill Book C°, 1929. 

Prasad, Ganesh. Some Great Mathematicians of the Nineteenth Century. Benares City: Benares Mathematical Society, 1933

Stewart, Ian. Significant Figures. New York: Hatchette Book Group, 2017.

Kolmogorov, A. N, and A. P. Yushkevich, eds. Mathematics of the 19th Century. Basel: Birkhäuser, 2001.