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More on imaginaries in p-adic fields

Published online by Cambridge University Press:  12 March 2014

Philip Scowcroft
Philip Scowcroft*
Affiliation:
Department of Mathematics, Wesleyan University, Middletown, CT 06459, USA, E-mail: pscowcroft@wesleyan.edu
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Extract

According to [4, p. 1154], a complete L-theory T eliminates imaginaries just in case for every L-formula φ(x1,… , xm, y1, …, yn), every model M of T, and every ā Є Mn, there is a subset A of M's domain with the following property: if N ≽ M and f is an automorphism of N, then

if and only if

Among the several equivalent conditions discussed in [4, p. 1155], one may single out the following: if T is a complete theory in which two distinct objects are definable, T eliminates imaginaries just in case every T-definable n-ary equivalence relation may be defined by a formula

where g is a T-definable n-ary function taking k-tuples as values (for some natural number k).

Say that an L-structure M eliminates imaginaries just in case Th(M) does. If L is the language of rings with unit, [4, p. 1158] shows that any algebraically closed field eliminates imaginaries, and [2, p. 629] points out that any real-closed field eliminates imaginaries.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 62 , Issue 1 , March 1997 , pp. 1 - 13
Copyright
Copyright © Association for Symbolic Logic 1997

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References

REFERENCES

[1]Bélair, L., Substructures and uniform elimination for p-adic fields, Annals of Pure and Applied Logic, vol. 39 (1988), pp. 1–17.CrossRefGoogle Scholar
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[6]Scowcroft, P. and Macintyre, A., On the elimination of imaginaries from certain valued fields, Annals of Pure and Applied Logic, vol. 61 (1993), pp. 241–276.CrossRefGoogle Scholar