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We show that any positive characteristic tame Hahn field$\mathbb{F}((t^\Gamma))$ containing $t$ is decidable in $\mathcal{L}_t$, thelanguage of valued fields with a constant symbol for $t$, if $\mathbb{F}$ and$\Gamma$ are decidable. In particular, we obtain decidability of$\mathbb{F}_p((t^{1/p^{\infty}}))$ and $\mathbb{F}_p((t^{\mathbb{Q}}))$ in$\mathcal{L}_t$. This uses a new AKE-principle for equal characteristic tamefields in $\mathcal{L}_t$, building on work by Kuhlmann, together withKedlaya's work on finite automata and algebraic extensions of function fields.In the process, we obtain an AKE-principle for tame fields in mixedcharacteristic.

Decidability of positive characteristic tame Hahn fields in $\mathcal{L}_t$