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# Undecidable wreath products and skew power series fields

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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 Xk((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.

## References

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[1]Ax, J., On the undecidability of power series fields, Proceedings of the American Mathematical Society, vol. 16 (1965), p. 846.
[2]Delon, F., Q muni de l'arithmetique faible de Penzin est décidable, to appear in Proceedings of the American Mathematical Society.
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[4]Delon, F. and Simonetta, P., Un principe d' Ax-Kochen-Ershov pour des structures intérmedials entre groupes et corps valués, to appear in this Journal.
[5]Neumann, B. H., On ordered division rings, Transactions of the American Mathematical Society vol. 66 (1949), pp. 202252.
[6]Neumann, B. H., On ordered groups, American Journal of Mathematics vol. 71 (1949), pp. 118.
[7]Neumann, B. H., Twisted wreath products of groups, Archiv der Mathematik vol. 17 (1963), pp. 16.
[8]Robinson, J., The decision problem for fields, Symposium on the theory of models, North-Holland, Amsterdam, 1965, pp. 299311.
[9]Robinson, R., Undecidable rings, Transactions of the American Mathematical Society vol. 70 (1951), pp. 137159.
[10]Simonetta, P., Décidabilité et interprétabilité dans les corps et les groupes non commutatifs, Thése, Université Paris 7, 1994.

# Undecidable wreath products and skew power series fields

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