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We study the relative algebraic closure $K$ of $\bar{\mathbb{F}}_p((t))$
inside $\bar{\mathbb{F}}((t^{\mathbb{Q}}))$. We show that the supports of
elements in $K$ have order type strictly less than $\omega^\omega$. 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 $\mathbb{F}_p((t^{\mathbb{Q}}))$, and other tame fields, in the
language of valued fields with a constant symbol for $t$.
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