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First edition, very rare, ANNOTATED BY KUMMER, of Kummer's first published work on factorizing certain types of complex numbers ('cyclotomic integers' in modern terminology), which was to prove crucial in the attempts by Kummer and others to prove Fermat's Last Theorem. In this paper, Kummer (1810-93) laid the foundations for the study of the cyclotomic integers, and indeed for algebraic number theory generally, such as the notions of conjugates of algebraic numbers, their norms, divisibility properties, prime elements, and units. There are two substantial marginal annotations by Kummer (pp. 8 & 24) and several minor corrections in ink in his hand. There are also several pencil annotations in a different hand, probably that of Jacob Lüroth (1844-1910). Every ordinary integer (whole number) can be expressed as a product of prime numbers, and this expression is unique up to re-ordering the factors. Kummer showed that, although every cyclotomic integer can be factorized as a product of primes, this factorization is not in general unique. Tradition has it that this work had been stimulated by Kummer's attempts to prove Fermat's Last Theorem, which states that the equation x^n + y^n = z^n, where n is an integer greater than 2, has no solution for positive integral values of x, y, and z. In 1843 Kummer showed Dirichlet an attempted proof of Fermat's Last Theorem, but Dirichlet pointed out that the proof depended on unique factorization. In the present paper, dedicated to the University of Königsberg on the occasion of its 300th Jubilaeum, Kummer showed, as Dirichlet had suspected, that this uniqueness property failed. This story is doubted by some, but in any case Kummer was not the only mathematician to fall into this trap. In March 1857, Gabriel Lamé delivered a lecture at the French Academy of Sciences in which he claimed to have a complete proof of Fermat's Last Theorem. Twelve weeks later, Lamé stood before the Academy and read a note from Kummer, who had pointed out that Lamé's proof depended on unique factorization, and that this property failed. Lamé translated the present paper by Kummer and published it next to his own in Liouville's Journal de Mathématiques Pures et Appliquées in 1847. But there was hope. Kummer showed in 1847 that unique factorization could be saved by using 'ideal complex numbers.' Kummer's ideal complex numbers would turn out to be a major breakthrough in the generalization of Fermat's Last Theorem. Using them Kummer was able to prove Fermat's Last Theorem in many cases, although not in general. "His contributions to Fermat's problem alone and his discovery of the basic ideas of algebraic number theory place him among the giants of mathematics" (Ribenboim, p. 1 in: Number Theory related to Fermat's Last Theorem, 1982). OCLC lists 3 copies in US. Library Hub records a single copy. No copies on RBH. 4to, pp. [vi], 28 (light browning, occasional minor stain, title page with single crease). Disbound.
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