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Let $K$ be the field $\mathbb{F_p}^{alg}((\mathbb{Q}))$ (field of Hahn series over $\mathbb{F_p}^{alg}$ and with value group $\mathbb{Q}$).

Is there elements $x$ of $K$ which are "almost prime" that is such that if $x=ab$ with $a, b\in K$, then $a=u(x^{1/p^k})^{p^k-n}$ and $b=u^{-1}(x^{1/p^k})^n$ where $u\in K$ ? And if so, how can such elements be recognized ?

Thank you to anyone who could help.

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