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Undecidable wreath products and skew power series fields

Published online by Cambridge University Press:  12 March 2014

Françoise Delon  and
Patrick Simonetta
Françoise Delon
Affiliation:
Équipe de Logique Mathématique de Paris VII, 2, Place Jussieu—Case 7012, 75251 Paris Cedex 05, France, E-mail: [email protected]
Patrick Simonetta
Affiliation:
Équipe de Logique Mathématique de Paris VII, 2, Place Jussieu—Case 7012, 75251 Paris Cedex 05, France, E-mail: [email protected]
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Extract

We prove the undecidability of a very large class of restricted and unrestricted wreath products (Theorem 1.2), and of some skew fields of power series (Section2). Both undecidabilities are obtained by interpreting some enrichments of twisted wreath products, which are themselves proved to be undecidable (Proposition 1.1).

We consider division rings of power series in various languages:

We show (Theorem 2.8) that every power series division ring k((B)), whose field of constants k is commutative and whose ordered group of exponents is noncommutative with a convex center, is undecidable in every extension of the language of rings where the valuation and the ordered group B are definable.

For certain k and B we prove here the undecidability of the structure

where X↾k((B))xB is the restriction of the multiplication to k((B)) Χ B,and γ is a given conjugation of k((B)). This shows that we cannot hope to improve our previous result, a sort of Ax-Kochen-Ershov principle for power series division rings, which ensures that

is decidable for every decidable solvable B.

Keywords

undecidable groupsundecidable fieldswreath productspower series division rings
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 63 , Issue 1 , March 1998 , pp. 237 - 246
Copyright
Copyright © Association for Symbolic Logic 1998

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References

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