Harmonic analysis as the exploitation of symmetry: A historical survey
George W. Mackey · History of mathematics · 2005
The first person to carry on the work begun by Lagrange and Legendre on the general theory of binary quadratic forms was C.F. Gauss (1777-1855).By his own account Gauss did not become aware of the work of his predecessors until he entered the university at Göttingen in the fall of 1795.Earlier the same year he accidentally discovered that if p is an odd prime, then -1 is a square mod p if and only if p is of the form An + 1.This excited him tremendously and, determined to get to the bottom of such phenomena, he had found and proved the quadratic reciprocity law by the end of March 1795.One thing led to another, and before arriving in Göttingen and beginning to study the works of Euler, Lagrange, and Legendre, Gauss had rediscovered many of their results.All this is explained in the introduction to his classic treatise Disquisitiones Arithmeticae published in 1801, which set the course for the future development of number theory.He claims that most of the material in the first four of the seven sections of his treatise was known to him before he arrived in Göttingen and that a large part of the book had been set up in type before Legendre's book of 1798 appeared.While he acknowledges that he was inspired to study quadratic forms by the work of Lagrange and Legendre, he reworked the whole subject in his own way, giving new proofs and introducing important new ideas and concepts.The seventh section contains Gauss's famous proof that for any prime/? of the form 2" + 1 (e.g., 17) one can give a ruler and compass construction of a regular polygon with p sides.He is reported to have definitely made up his mind to be a mathematician when he found this beautiful result in the spring of 1796.At this point the celebrated Galois theory of equations was thirty-five years in the future, but Gauss's proof is imbedded in what amounts to a complete development of that theory for the equation x p -i = o.(x p -1 factors into JC -1 and an irreducible polynomial of degree/?-1.If p -1 = 2\ it follows from Galois theory that the equation can be solved by rational operations and the taking of square roots.)In his work on the theory of binary quadratic forms, Gauss confined himself to the case in which the middle coefficient is even, writing Ax 2 + IBxy + Cy 2 and defining the discriminant to be B 2 -AC instead of B 2 -4A C. Also he pointed out that simplifications ensue if one makes a distinction between "proper" and "improper" equivalence of forms; two forms being said to be properly equivalent only when the transformation matrix (S) has determinant 1 rather than -1.By far his most important contribution, however, was his observation that there is a natural (but not obvious) composition law for proper equivalence classes of forms having a fixed discriminant and that this composition law converts the finite set of classes with square free discriminant D into a finite commutative group.Of