MATHEMATICS IN THE NINETEENTH CENTURY 485 



Another part of the theory of numbers also goes back to Gauss, 

 namely, algebraic numerical bodies. The Law of Reciprocity of 

 Quadratic Residues, one of the gems of the higher arithmetic, was 

 first rigorously proved by Gauss. His attempts to extend this 

 theorem to cubic and biquadratic residues showed that the elegant 

 simplicity which prevailed in quadratic residues was altogether 

 missing in these higher residues, until one passed from the domain 

 of real integers to the domain formed of the third and fourth roots of 

 unity. In these domains, as Gauss remarked, algebraic integers have 

 essentially the same properties as ordinary integers. Further explor- 

 ation in this new and promising field by Jacobi, Eisenstein, and 

 others soon brought to light the fact that already in the domain 

 formed of the twenty-third roots of unity the laws of divisibility were 

 altogether different from those of ordinary integers; in particular, 

 a number could be expressed as the product of prime factors in more 

 than one way. Further progress in this direction was therefore 

 apparently impossible. 



It is Kummer's immortal achievement to make further progress 

 possible by the invention of his ideals. These he applied to Fermat's 

 celebrated Last Theorem and the Law of Reciprocity of Higher 

 Residues. 



The next step in this direction was taken by Dedekind and Kro- 

 necker, who developed the ideal theory for any algebraic domain. 

 So arose the theory of algebraic numerical bodies, which has come 

 into such prominence in the last decades of the century through 

 the researches of Hensel, Hurwitz, Minkowski, Weber, and, above 

 all, Hilbert. 



Kronecker has gone farther, and in his classic Grundzuge he has 

 shown that similar ideas and methods enable us to develop a theory 

 of algebraic bodies in any number of variables. The notion of divis- 

 ibility so important in the preceding theories is generalized by Kro- 

 necker still farther in the shape of his system of moduli. 



Another noteworthy field of research opened up by Kronecker is 

 the relation between quadratic forms with negative determinant 

 and complex multiplication of elliptic functions. H. Smith, Gierster, 

 Hurwitz, and especially Weber have made important contributions. 

 A method of great power in certain investigations has been created 

 by Minkowski, which he called the Geometric der Zahlen. Introduc- 

 ing a generalization of the distance function, he is led to the concep- 

 tion of a fundamental body (Aichkorper') . Minkowski shows that 

 every fundamental body is nowhere concave, and conversely to 

 each such body belongs a distance function. A theorem of great 

 importance is now the following: The minimum value which each 

 distance function has at the lattice points is not greater than a certain 

 number depending on the function chosen. 



