Submission date: 15 January 2023

Abstract:

We study the relative algebraic closure K of 𝔽_p((t)) inside 𝔽_p((t^ℚ)). We show that the supports of elements in K have order type strictly less than ω^ω. We also recover a theorem by Rayner giving a bound to the ramification away from p in the support of elements in K, and an analogue of Rayner's result for the residue field. This work has applications to the decidability of the first order theory of 𝔽_p((t^ℚ)), and other tame fields, in the language of valued fields with a constant symbol for t.

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Full text arXiv 2301.06177: pdf, ps.