Gauss, Dirichlet, and the Law of Biquadratic Reciprocity

Gauss and Dirichlet are two of the most influential figures in the history of number theory. Gauss’s monumental Disquisitiones Arithmeticae, first published in 1801, synthesized many earlier results and served as a point of departure for the modern approach to the subject (Goldstein et al. 2007). The three principal sections of the book were devoted to the theory of congruences (where Gauss introduced the still standard notation a ≡ b(mod m)), the classical subject of quadratic forms that had been studied by Fermat, Euler, Lagrange, and even Diophantos, and the theory of cyclotomic equations. Although the appearance of this masterpiece did much to establish Gauss’s early mathematical fame, the sheer novelty of the work together with its rigidly formal exposition made it difficult for all but the greatest of his contemporaries to appreciate it fully. One of those who did was Lagrange, the aged giant who, along with Euler, stood at the pinnacle of eighteenth-century mathematics. In 1804, he wrote the young protégé of the Duke of Brunswick, “With your Disquisitiones you have at once arrayed yourself among the mathematicians of the first rank, and I see that your last section on cyclotomic equations contains the most beautiful analytic discoveries that have been made for a long time.” (Wussing 1974, 32) (Fig. 3.1).