  We show that any positive characteristic tame Hahn field
$\mathbb{F}((t^\Gamma))$ containing $t$ is decidable in $\mathcal{L}_t$, the
language 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 tame
fields in $\mathcal{L}_t$, 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.
