Why Dickson Left Quadratic Reciprocity Out of His History of the Theory of Numbers

Abstract

In a 1993 letter to the Notices of the American Mathematical Society, Irving Kaplansky called attention to an astonishing omission in the history of mathematics [27]. Everybody knows, Kaplansky asserted, Dickson's History of the Theory of Numbers covers all of number up to about 1918. Right? Wrong, answered solidly, [t]ry looking up quadratic reciprocity. Kaplansky is right. Leonard Eugene Dickson's monumental compendium of the history of number excludes the history of quadratic reciprocity, the crown jewel of elementary number theory [27]. Why? Why did Dickson leave this celebrated number theoretic result out of his History? Kaplansky offers a brief explanation: he farmed out to a student [27]. Again, Kaplansky is right, on some level at least. In this paper, we offer further insight into this perplexing omission. In the process, we reveal an entirely new perspective on Dickson and unfold yet another example in the history of mathematics where extra-mathematical factors contribute to the development of mathematics. The history of Dickson's History actually begins in the last decade of the nineteenth century. While Dickson pursued a Ph.D. at the young Chicago from 1894 to 1896, the then group-theoretically minded E. H. Moore inspired him to write a thesis on (what we would call) permutation groups [15]. Although group would remain among Dickson's research interests throughout his career, would add finite field theory, invariant theory, the of algebras, and number to his repertoire of research interests. In the spring of 1900, just a few months past his twenty-sixth birthday, the Chicago Mathematics Department invited Dickson to join them as an assistant professor. From this position, Dickson made significant contributions to the consolidation and growth of the algebraic tradition in America [23]. Specifically, Dickson spent forty years (all but the first two) of his professional career on the faculty at Chicago where directed 67 Ph.D. students, wrote 18 books and roughly 300 manuscripts, served as editor of the American Mathematical Monthly and the Transactions of the American Mathematical Society, and guided the American Mathematical Society as its President from 1916 to 1918 [3]. Yet, this mathematical workhorse, who played billiards and bridge by day and did mathematics from 8:30 to 1:30 a.m. every night [1, 377], interrupted his thriving pure mathematical career for nearly a decade to write a three-volume, 1500 page historical account of the of numbers. As explained himself, undertook this project because it fitted with my conviction that every person should aim to perform at some time in his life some serious useful work for which