Decidability of Laurent series fields F_q((t)) in the language of rings

Determine whether, for each finite field F_q, the first-order theory Th(F_q((t))) in the language of rings is decidable. This asks for an algorithm that decides the truth of every first-order L_ring sentence in the Laurent series field F_q((t)).

Background

Ax and Kochen established the decidability of the first-order theory of p-adic fields, but the analogous question for Laurent series fields over finite fields has resisted resolution. Despite substantial progress on fragments—most notably the proof by Anscombe and Fehm that the existential theory of F_q((t)) is decidable—the decidability of the full first-order theory remains unsettled.

The present paper contributes to the broader landscape by showing existential undecidability for certain expansions of F_q((t)) by naming an element of positive valuation and a cyclic multiplicative subgroup it generates. These results highlight the subtlety of definability and decidability phenomena around F_q((t)), but they do not resolve the core question of whether Th(F_q((t))) itself is decidable.