Abel, Galois, Klein, Noether: Unsolvability, Symmetry, and Unity in Mathematics in the 19th and 20th Centuries

Course Description

Summary

What does it mean for a mathematical problem to be unsolvable? The very concept does not seem to have been much considered, until, in 1824, a young Norwegian named Niels Henrik Abel published a small pamphlet on an old problem. The pamphlet was one of the first markers of a sea change in mathematics, and by the time Abel died, six years later at the age of twenty-six, mathematics was becoming a different place. 

The solution of quadratic polynomial equations has been taught in school, on and off, since the time of the Babylonians. Attempts to solve cubic polynomial equations go back through history, with a general method for any cubic equation being discovered in the sixteenth century; the solution of the quartic polynomial equation followed shortly thereafter. But the solution of the quintic polynomial equation, in terms of algebraic operations and radicals, proved stubborn, despite many clever strategies and partial answers. This was the problem that the young Abel proved impossible to solve.

Other impossibility proofs soon followed: the ancient problems of using a compass and straightedge to trisect an angle, double a cube, or square a circle, were all proved impossible to solve. The idea of limitation had begun to take hold. Even more profoundly, new intellectual horizons were opened up. In 1832, the night before he was to die in a duel, the twenty year old Évariste Galois sketched in a letter his conceptions of how the unsolvability of polynomial equations was bound up with questions of symmetry. These questions of symmetry came to pervade mathematics, providing the underpinning for Felix Klein's program of unification of mathematics in the late nineteenth century, and taking a new, elevated form in the twentieth century in the visionary work of Emmy Noether. The new ideas reached backwards as well: Klein showed how the quintic could be made solvable, if one admitted new types of complex functions, rooted in the symmetry of the icosahedron. The concepts continue to be central, from error-correcting codes, to a periodic table for elementary particles that underlies the structure of matter ("The Eightfold Way").

This class is not primarily a history class; the main focus is the mathematics, embedded in the historical narrative. We follow the original motivations, and the story of the mathematics as it was developed, in particular reading Abel's and Galois' original papers. However, we will simplify, and jump forward in time as needed, in order to make the mathematics more clear. The mathematics is still very contemporary; it is only the approach that is historically oriented.

Students may take this class at different levels of mathematical intensity. If you complete all mathematical details in every assignment, this will be a quite rigorous advanced level mathematics class. If you complete only the core mathematical elements, this can be a mathematics class at the introductory level. Or, taking a middle ground, this can also be an intermediate level mathematics class. Also, the amount of history that comes into the work is up to the individual student: though it is not a history class, a student could choose to take the class with a strong historical focus. What each student ends up completing will be explained in the narrative evaluation. The final evaluation will be relative to the student goals. For example, a student can get an "A" if they choose to take this as an introductory class, doing only the core and basic ideas, and doing a good job with those; the evaluation would identify that the student took it as an introductory class.

The only formal prerequisite is a comfort with high school polynomial algebra (e.g. expanding, factoring). To do more advanced assignments, having taken MAT 4288 Linear Algebra: An Introduction, or a similar course, and having taken a college mathematics course involving proofs, will be helpful.

The overall scope of the course is similar to the book Abel's Proof: An Essay on the Sources and Meaning of Mathematical Unsolvability, by Peter Pesic. References for the more advanced material are Harold Edwards, Galois Theory, and Felix Klein, The Icosahedron and the Solution of Equations of the Fifth Degree. Taking this course at the intermediate or advanced level counts as a one semester class in Abstract Algebra.