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Paper   IPM / M / 7814
School of Mathematics
  Title:   Real closed fields and IP-sensitivity
1.  S. M. Ayat
2.  Moj. Moniri
  Status:   Published
  Journal: Lect. Notes Log.
  Vol.:  26
  Year:  2006
  Pages:   1-22
  Supported by:  IPM
This work is concerned with the extent to which certain topological and algebraic phenomena, such as the mod one density of the square roots of positive integers, or the existence of irrationals and transcendentals, generalize from the standard setting of 〈\mathbbR, \mathbbZ〉 to structures of the form 〈F,I〉, where F is an ordered field and I is an integer part (IP) for F. Here are some of the highlights of the paper:
(i) Given an ordered field F, either all or none of the IP's for F (if any) are models of Open Induction, depending on whether F is dense in its real closure or not.
(ii) Density mod one of a subset is sensitive to a certain relaxation of the notion of an IP in non-Archimedean real closed fields with countable IP's. The weakened notion is that of (additive subgroup) integer sets in the sense of J. Schmerl.
(iii) There is a real closed field F with two IP's I1 and I2 such that for some α ∈ I2 which is an algebraic irrational with respect to I1, the set {uα−⎣uα⎦I1|uI1} is dense in [0,1)F.
(iv) All real closed fields F of infinite transcendence degree have proper dense real closed subfields over which F can be of any positive transcendence degree, up to and including that of the transcendence degree of F (they are real closures of suitable proper dense subfields also). Together with a result of van den Dries on the existence of real closed fields F of infinite transcendence degree which possess certain IP's I with the peculiar property that every element of F is rational with respect to I, this demonstrates the IP-sensitivity of existence of transcendentals in F.
(v) Certain classes of real closed fields of infinite transcendence degree such as Puiseux series, and all real closed fields of finite transcendence degree, have algebraic irrationals relative to every IP. Also, Cauchy complete real closed fields have transcendentals relative to any IP.
(vi) For all n ∈ \mathbbN ≥ 1, there are real closed fields F and K of transcendence degree n such that F is algebraic over all its proper dense subfields, but K is algebraic over a proper dense subfield of itself, but transcendental over another proper dense subfield of itself. This and the previous two issues are also dealt with in the second paper of F.-V. Kuhlmann in the present volume.

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