ALGEBRAIC NU几BERB 819 formulated."This particular case of n=7 was disposed of by Lame in 1839,10 and Dirichlet established the assertion for n=14.11 However,the general proposition was unproven. It was taken up by Ernst Eduard Kummer(1810-93),who turned from theology to mathematics,became a pupil of Gauss and Dirichlet,and later served as a professor at Breslau and Berlin.Though Kummer's major work was in the theory of numbers,he made beautiful discoveries in geometry which had their origin in optical problems;he also made important con- tributions to the study of refraction of light by the atmosphere. Kummer took x+y where is prime,and factored it into (x+)(x+g）.(+a-y) where a is an imaginary pth root of unity.That is,a is a root of (1) a-1+a2-2+.+a+1=0, This led him to extend Gauss's theory of complex integers to algebraic numbers insofar as they are introduced by cquations such as(1),that is, numbers of the form f八a）=a0+41a+.+ap-2a-3 where each a is an ordinary (rational)integer.(Since a satisfies (1),terms in a-1 can be replaced by terms of lower power.)Kummer called the numbers f(a)complex integers. By 1843 Kummer made appropriate definitions of integer,prime integer,divisibility,and the like(we shall give the standard definitions in a moment)and then made the mistake of assuming that unique factorization holds in the class of algebraic numbers that he had introduced.He pointed out while transmitting his manuscript to Dirichlet in 1843 that this assump- tion was necessary to pr ove Fermat's theorem.Dirichlet informed him tha unique factorization holds only for certain primes A.Incidentally,Cauchy and Lame made the same mistake of assuming unique factorization for algebraic numbers.In 1844 Kummer1a recognized the correctness of Dirichlet's criticism. To restore unique factorization Kummer created a theory of ideal numbers in a series of papers starting in 1844.13 To understand his idea let us consider the domain of a +bv-5,where a and b are integers.In this domain 6=2.3=(1+V-5(1-V-5 10.Jow.de Math.,5,1840,195-211. 11.Jo.far Math.,9,1832,390-93=Wrk,1,18994 12.Jota.de Math.,12,1847,185-212. 13.Jow.far Math.,35,1847,319-26,327-67