Editing Ideal number (section)

==History==
Kummer first published the failure of unique factorization in [[cyclotomic field]]s in 1844 in an obscure journal; it was reprinted in 1847 in [[Joseph Liouville|Liouville's]] journal. In subsequent papers in 1846 and 1847 he published his main theorem, the unique factorization into (actual and ideal) primes.

It is widely believed that Kummer was led to his "ideal [[complex number]]s" by his interest in [[Fermat's Last Theorem]]; there is even a story often told that Kummer, like [[Gabriel Lamé|Lamé]], believed he had proven Fermat's Last Theorem until [[Peter Gustav Lejeune Dirichlet|Lejeune Dirichlet]] told him his argument relied on unique factorization; but the story was first told by [[Kurt Hensel]] in 1910 and the evidence indicates it likely derives from a confusion by one of Hensel's sources. [[Harold Edwards (mathematician)|Harold Edwards]] says the belief that Kummer was mainly interested in Fermat's Last Theorem "is surely mistaken" (Edwards 1977, p. 79). Kummer's use of the letter λ to represent a prime number, α to denote a λth root of unity, and his study of the factorization of prime number <math>p\equiv 1 \pmod{\lambda}</math> into "complex numbers composed of <math>\lambda</math>th roots of unity" all derive directly from a paper of [[Carl Gustav Jakob Jacobi|Jacobi]] which is concerned with [[reciprocity law|higher reciprocity laws]]. Kummer's 1844 memoir was in honor of the jubilee celebration of the University of Königsberg and was meant as a tribute to Jacobi. Although Kummer had studied Fermat's Last Theorem in the 1830s and was probably aware that his theory would have implications for its study, it is more likely that the subject of Jacobi's (and [[Carl Friedrich Gauss|Gauss's]]) interest, higher reciprocity laws, held more importance for him. Kummer referred to his own partial proof of Fermat's Last Theorem for [[regular prime]]s as "a curiosity of number theory rather than a major item" and to the higher reciprocity law (which he stated as a conjecture) as "the principal subject and the pinnacle of contemporary number theory." On the other hand, this latter pronouncement was made when Kummer was still excited about the success of his work on reciprocity and when his work on Fermat's Last Theorem was running out of steam, so it may perhaps be taken with some skepticism.

The extension of Kummer's ideas to the general case was accomplished independently by Kronecker and Dedekind during the next forty years. A direct generalization encountered formidable difficulties, and it eventually led Dedekind to the creation of the theory of [[module (mathematics)|modules]] and [[ideal (ring theory)|ideals]]. Kronecker dealt with the difficulties by developing a theory of forms (a generalization of [[quadratic forms]]) and a theory of [[divisor (algebraic geometry)|divisors]]. Dedekind's contribution would become the basis of [[ring theory]] and [[abstract algebra]], while Kronecker's would become major tools in [[algebraic geometry]].