Submission date: 9 August 2021

Abstract:

We show that any positive characteristic tame Hahn field 𝔽((t^Γ)) containing t is decidable in L_t, the language of valued fields with a constant symbol for t, if 𝔽 and Γ are decidable. In particular, we obtain decidability of 𝔽_p((t^{1/p^{∞}})) and 𝔽_p((t^ℚ)) in L_t. This uses a new AKE-principle for equal characteristic tame fields in L_t, building on work by Kuhlmann, together with Kedlaya's work on finite automata and algebraic extensions of function fields. In the process, we obtain an AKE-principle for tame fields in mixed characteristic and recover a theorem by Rayner on the relative algebraic closure of function fields inside Hahn fields.

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Full text arXiv 2108.04132: pdf, ps.