Diophantine undecidability in some rings of algebraic numbers of totally real infinite extensions of \(\mathbb{Q}\)

DOI10.1016/0168-0072(94)90024-8zbMath0816.11066OpenAlexW2070490082MaRDI QIDQ1331916

Alexandra Shlapentokh

Publication date: 29 August 1994

Published in: Annals of Pure and Applied Logic (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1016/0168-0072(94)90024-8

zbMATH Keywords

analogue of Hilbert's tenth problemdiophantine equationpositive answerring of all algebraic numbers

Mathematics Subject Classification ID

Decidability (number-theoretic aspects) (11U05) Decidability of theories and sets of sentences (03B25) Diophantine equations (11D99)

Related Items (10)

Diophantine definability and decidability in extensions of degree 2 of totally real fields ⋮ First-order decidability and definability of integers in infinite algebraic extensions of the rational numbers ⋮ Definability of the natural numbers in totally real towers of nested square roots ⋮ Existential definability and diophantine stability ⋮ Diophantine undecidability for some function fields of infinite transcendence degree and positive characteristic ⋮ Extensions of Hilbert’s Tenth Problem: Definability and Decidability in Number Theory ⋮ Rings of algebraic numbers in infinite extensions of \(\mathbb Q\) and elliptic curves retaining their rank ⋮ Diophantine undecidability of holomorphy rings of function fields of characteristic 0 ⋮ DIOPHANTINE DEFINABILITY OVER HOLOMORPHY RINGS OF ALGEBRAIC FUNCTION FIELDS WITH INFINITE NUMBER OF PRIMES ALLOWED AS POLES ⋮ On Dipphantine definability and decidability in some rings of algebraic functions of characteristic 0

Cites Work

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• A Diophantine definition of rational integers over some rings of algebraic numbers
• Extension of Hilbert's tenth problem to some algebraic number fields
• Hilbert's Tenth Problem for a Class of Rings of Algebraic Integers
• Diophantine relationships between algebraic number fields
• Arithmetic over the rings of all algebraic integers.
• Diophantine Sets Over Algebraic Integer Rings. II
• A diophantine definition of integers in the rings of rational numbers
• Hilbert's Tenth Problem for Quadratic Rings
• Diophantine Sets over Some Rings of Algebraic Integers
• Hilbert's Tenth Problem is Unsolvable

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