| Mathematician John Swallow recently published a book on Galois theory so that undergraduates wouldn’t find it as inscrutable as he did.
An active researcher, he has established himself as a leading scholar in the field. But when he first encountered Galois theory as a Yale graduate student fourteen years ago, was overwhelmed by the degree of abstraction. “It would have been so much better to see concrete examples the first time around,” he said. “Instead, I learned powerful theorems without really knowing what made them so powerful.”
Evariste Galois was a brilliant young French mathematician of the early 19th century, and his point of view on roots of polynomials inspired the theory that now bears his name. Although he was killed in a duel in 1832 at the age of twenty, mathematicians have long valued his ideas for solving algebraic problems.
In fact, they generalized Galois’ approach to settings far removed from its origins. Unfortunately for students, textbooks followed the trend. “Seeing ideas from Galois theory across so many areas of algebra makes for a great story,” Swallow said, “unless that’s the story you’re trying to introduce to beginners.”
Swallow, the Kimbrough associate professor of mathematics, wrote Exploratory Galois Theory to offer undergraduates an introduction to its concepts, theorems, and proofs at an extremely accessible level. The book approaches the theory in small bits, offering exercises with hints that encourage students to complete proofs. The polynomial roots in the book are concrete rational numbers, square roots, cube roots and so on. The book avoids numbers Swallow says are “visible only to graduate students.” “The text focuses on numbers we encounter in high school,” he explained, “and just leaves the rest for another day.”
Earlier texts had abandoned a concrete approach because the computations required for specific examples were so tedious. But with the increasing power of computing, Galois theorists gained the ability to perform calculations and understand examples that had been inaccessible with pencil and paper.
The increasing speed of desktop computers in the past several years allowed Swallow to include another feature in his project. He wrote plug-in modules for two popular mathematical software applications, Maple and Mathematica, that give students a set of user-friendly tools for experimentation with polynomial roots. His book explains how to use these packages, called “AlgFields,” and the source code for the packages is freely available to interested students and faculty under the GNU public license.
Many undergraduate institutions don’t offer courses in Galois theory, and Swallow hopes his book will provide professors a resource to give Galois theory another chance. At schools that do offer it, the courses are typically aimed at junior and senior mathematics majors who’ve studied calculus and abstract algebra, and often are considering graduate study in math.
Swallow has taught Galois theory semi-annually at Davidson since 1998, using developing drafts of his book. Following a mathematical tradition, he enlisted his students’ help in improving the drafts, offering them dollar for every mistake they found. He confesses to paying out about $30 of such rewards, and considers it a great investment for the 224-page textbook.
Swallow has enjoyed writing since double-majoring in English literature and mathematics as an undergraduate at Sewanee, the University of the South. Once at Davidson, he reflected on his passion for both areas in an essay, “Beautiful Numbers,” published in 1995 in Phi Beta Kappa’s journal The American Scholar. The article describes his quest to satisfy his aesthetic sense in mathematics in graduate school.
After enrolling at Yale, he took mandatory exams in analysis, topology, and algebra. He thought he understood algebra better than the other two, but he scored the lowest on the algebra exam. Reviewing his performance, he realized that proofs of algebraic theorems are sometimes deeper and more compelling than they seem at first. As he wrote in his essay, “I recognized that elegant and beautiful statements in algebra, while expressing seemingly simple facts, require surprisingly subtle arguments in their proofs. I sensed a real depth to the subject and decided that’s what I wanted to participate in.”
Swallow’s aesthetic sense isn’t confined to mathematics. For the past five years, he has taught a section in the college’s Humanities Program, exploring the Western Tradition. He spent his first sabbatical in 1998-1999 in a large mathematics department in a large university, but he found the atmosphere too rarified. “It was a wonderful year with algebraists and friends, but I’m much more at home in a small-college environment,” he said. “At Davidson I can teach a course a year in Humanities and work with faculty from across the college. To me, it’s very satisfying.”
Davidson is a highly selective independent liberal arts college for 1,600 students. Since its establishment in 1837, the college has graduated 23 Rhodes Scholars and is consistently ranked in the top ten liberal arts colleges in the country by U.S. News and World Report magazine. Davidson is engaged in “Let Learning Be Cherished,” a $250 million campaign in support of student financial assistance, academic resources, and community life.
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