Hilbert's tenth problem for fields of rational functions over finite fields
type: Item
instance of: scholarly article
title: Hilbert's tenth problem for fields of rational functions over finite fields
zbMATH Open document ID: 0696.12022
DOI: 10.1007/BF01239506
published in: Inventiones Mathematicae
publication date: 1991-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143850
review text: Let F be a finite field of characteristic \(p>2\) and F(t) the field of rational functions in the variable t with coefficients in F. It is proved that the existential theory of F(t) in the language \(\{+,\cdot,0,1,t\}\) is undecidable, and so, there is no algorithm to solve arbitrary polynomial equations over F(t); this is an analogue to Hilbert's tenth problem for the case of a function field. Due to the similarities between \({\mathbb{Q}}\) (the field of rationals) and the common properties of function fields over finite fields, when it comes to the solvability of diophantine equations, this result seems to suggest that the analogue of Hilbert's tenth problem for \({\mathbb{Q}}\) has a negative answer.
Mathematics Subject Classification ID: 11U05
Mathematics Subject Classification ID: 11R58
Mathematics Subject Classification ID: 11D99
zbMATH DE Number: 4139893
zbMATH Keywords: polynomial equations over field of rational functions
zbMATH Keywords: Hilbert's tenth problem
zbMATH Keywords: function fields over finite fields
author: Thanases Pheidas
reviewed by: Thanases Pheidas
MaRDI profile type: Publication
cites work: Q3344192
cites work: Q4115143
cites work: Q3905379
cites work: The Diophantine Problem for Polynomial Rings and Fields of Rational Functions
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description: scientific article; zbMATH DE number 4139893