|Abstract||By the middle of the nineteenth century, it had become clear to mathematicians that the study of ﬁnite ﬁeld extensions of the rational numbers is indispensable to number theory, even if one’s ultimate goal is to understand properties of diophantine expressions and equations in the ordinary integers. It can happen, however, that the “integers” in such extensions fail to satisfy unique factorization, a property that is central to reasoning about the ordinary integers. In 1844, Ernst Kummer observed that unique factorization fails for the cyclotomic integers with exponent 23, i.e. the ring Z[ζ] of integers of the ﬁeld Q(ζ), where ζ is a primitive twenty-third root of unity. In 1847, he published his theory of “ideal divisors” for cyclotomic integers with prime exponent. This was to remedy the situation by introducing, for each such ring of integers, an enlarged domain of divisors, and showing that each integer factors uniquely as a product of these. He did not actually construct these integers, but, rather, showed how one could characterize their behavior qua divisibility in terms of ordinary operations on the associated ring of integers.|
|Keywords||No keywords specified (fix it)|
No categories specified
(categorize this paper)
|Through your library||Only published papers are available at libraries|
Similar books and articles
Stuart T. Smith (1992). Prime Numbers and Factorization in IE1 and Weaker Systems. Journal of Symbolic Logic 57 (3):1057 - 1085.
Alexandra Shlapentokh (2011). Defining Integers. Bulletin of Symbolic Logic 17 (2):230-251.
Vladimir Kanovei (1996). On External Scott Algebras in Nonstandard Models of Peano Arithmetic. Journal of Symbolic Logic 61 (2):586-607.
Denis Richard (1985). Answer to a Problem Raised by J. Robinson: The Arithmetic of Positive or Negative Integers is Definable From Successor and Divisibility. Journal of Symbolic Logic 50 (4):927-935.
Patrick Cegielski, Yuri Matiyasevich & Denis Richard (1996). Definability and Decidability Issues in Extensions of the Integers with the Divisibility Predicate. Journal of Symbolic Logic 61 (2):515-540.
Harvey Friedman, The Number of Certain Integral Polynomials and Nonrecursive Sets of Integers, Part.
George Weaver (2011). A General Setting for Dedekind's Axiomatization of the Positive Integers. History and Philosophy of Logic 32 (4):375-398.
Stewart Shapiro (2000). Frege Meets Dedekind: A Neologicist Treatment of Real Analysis. Notre Dame Journal of Formal Logic 41 (4):335--364.
Added to index2009-01-28
Total downloads5 ( #169,941 of 722,813 )
Recent downloads (6 months)0
How can I increase my downloads?