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Weak presentations of computable fields

Published online by Cambridge University Press:  12 March 2014

Carl G. Jockusch Jr  and
Alexandra Shlapentokh
Carl G. Jockusch Jr
Affiliation:
Department of Mathematics, University of Illinois, 1409 West Green Street, Urbana, Illinois 61801, E-mail: jockusch@math.uiuc.edu
Alexandra Shlapentokh
Affiliation:
Department of Mathematics, East Carolina University, Greenville, NC 27858, E-mail: mashlape@ecuvax.cis.ecu.edu
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Abstract

It is shown that for any computable field K and any r.e. degree a there is an r.e. set A of degree a and a field F ≃ K with underlying set A such that the field operations of F (including subtraction and division) are extendible to (total) recursive functions. Further, it is shown that if a and b are r.e. degrees with b ≤ a, there is a 1-1 recursive function f: ℚ → ω such that f(ℚ) ∈ a, f(ℤ) ∈ b, and the images of the field operations of ℚ under f can be extended to recursive functions.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 60 , Issue 1 , March 1995 , pp. 199 - 208
Copyright
Copyright © Association for Symbolic Logic 1995

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References

REFERENCES

[1]Fried, M. and Jarden, M., Field arithmetic, Springer-Verlag, Berlin, 1986.CrossRefGoogle Scholar
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