Page:A History of Mathematics (1893).djvu/390

 $$\scriptstyle{x^2-1=0}$$ and complex numbers $$\scriptstyle{a+b\rho}$$ ($\scriptstyle{\rho}$ being a cube root of unity), the theory of which resembles that of Gauss' numbers. Kummer passed to the general case $$\scriptstyle{x^n-1=0}$$ and got complex numbers of the form $\scriptstyle{\alpha=\alpha_1A_1+\alpha_2A_2+\alpha_3A_3+\cdots}$, where $$\scriptstyle{\alpha_i}$$ are whole real numbers, and $$\scriptstyle{A_i}$$ roots of the above equation.[59] Euclid's theory of the greatest common divisor is not applicable to such complex numbers, and their prime factors cannot be defined in the same way as prime factors of common integers are defined. In the effort to overcome this difficulty, Kummer was led to introduce the conception of "ideal numbers." These ideal numbers have been applied by G. Zolotareff of St Petersburg to the solution of a problem of the integral calculus, left unfinished by Abel (Liouville's Journal, Second Series, 1864, Vol. IX.). Julius Wilhelm Richard Dedekind of Braunschweig (born 1831) has given in the second edition of Dirichlet's Vorlesungen über Zahlentheorie a new theory of complex numbers, in which he to some extent deviates from the course of Kummer, and avoids the use of ideal numbers. Dedekind has taken the roots of any irreducible equation with integral coefficients as the units for his complex numbers. Attracted by Kummer's investigations, his pupil, Leopold Kronecker (1823–1891) made researches which he applied to algebraic equations.

On the other hand, efforts have been made to utilise in the theory of numbers the results of the modern higher algebra. Following up researches of Hermite, Paul Bachmann of Münster investigated the arithmetical formula which gives the automorphics of a ternary quadratic form.[89] The problem of the equivalence of two positive or definite ternary quadratic forms was solved by L. Seeber; and that of the arithmetical automorphics of such forms, by Eisenstein. The more difficult problem of the equivalence for indefinite ternary forms has been investigated by Edward Selling of Würzburg. On quadratic