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Undecidability of rational function fields in nonzero characteristic. (English) Zbl 0551.03027
Logic colloquium ’82, Proc. Colloq., Florence 1982, Stud. Logic Found. Math. 112, 85-95 (1984).
[For the entire collection see Zbl 0538.00003.]
Let F be an infinite perfect field of characteristic \(p>0\). The author proves the undecidability of F(t), the rational function field, in the pure language of fields. In his proof the author first shows that certain weak monadic theories of such F are undecidable. He then shows how these theories can be interpreted in F(t). An undecidability result is also proven for F((t)), the formal power series field, in the language of valued fields with an additional predicate for F and the cross section function. The paper ends with a discussion of open problems.
Reviewer: J.M.Plotkin

MSC:
03D35 Undecidability and degrees of sets of sentences
03B25 Decidability of theories and sets of sentences