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)|
|Categories||categorize this paper)|
References found in this work BETA
No references found.
Citations of this work BETA
No citations found.
Similar books and articles
Prime Numbers and Factorization in IE1 and Weaker Systems.Stuart T. Smith - 1992 - Journal of Symbolic Logic 57 (3):1057 - 1085.
On External Scott Algebras in Nonstandard Models of Peano Arithmetic.Vladimir Kanovei - 1996 - Journal of Symbolic Logic 61 (2):586-607.
Answer to a Problem Raised by J. Robinson: The Arithmetic of Positive or Negative Integers is Definable From Successor and Divisibility.Denis Richard - 1985 - Journal of Symbolic Logic 50 (4):927-935.
Definability and Decidability Issues in Extensions of the Integers with the Divisibility Predicate.Patrick Cegielski, Yuri Matiyasevich & Denis Richard - 1996 - Journal of Symbolic Logic 61 (2):515-540.
The Number of Certain Integral Polynomials and Nonrecursive Sets of Integers, Part.Harvey Friedman - manuscript
A General Setting for Dedekind's Axiomatization of the Positive Integers.George Weaver - 2011 - History and Philosophy of Logic 32 (4):375-398.
Frege Meets Dedekind: A Neologicist Treatment of Real Analysis.Stewart Shapiro - 2000 - Notre Dame Journal of Formal Logic 41 (4):335--364.
Added to index2009-01-28
Total downloads32 ( #161,319 of 2,172,604 )
Recent downloads (6 months)1 ( #325,028 of 2,172,604 )
How can I increase my downloads?