Breakthrough Tracker record
Hilbert's tenth problem for rings of integers of number fields
Two independent teams prove that no algorithm can always decide whether an arbitrary polynomial equation has a solution in the ring of integers of a given number field.
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- Stable ID
math-hilbert-tenth-number-fields-2024- Revision
math-hilbert-tenth-number-fields-2024.v1- Field
- Mathematics · Number theory and logic
- Evidence
- Tier 1 · Peer reviewed: Yes
- Record state
- Current · One proof published; independent proof forthcoming
- Last checked
AI role
No generative-AI role was disclosed.
Record details
- Problem or result
- Hilbert's tenth problem over rings of integers of number fields
- Authors
- Peter Koymans and Carlo Pagano; independently Levent Alpöge, Manjul Bhargava, Wei Ho and Ari Shnidman
- Institutions
- Utrecht University and Concordia University for the first team; see the second manuscript for its exact affiliations
- Result date
- One proof published online December 1, 2025, in a March 2026 volume; independent proof forthcoming in JAMS
Why it matters
The work extends the negative solution of Hilbert's tenth problem from ordinary integers to the rings of integers of all number fields.
Limits
The quantified domain is rings of integers of number fields. It should not be broadened casually to every ring described as involving algebraic integers.
Sources
- Primary: Koymans and Pagano, Hilbert's tenth problem via additive combinatorics
- Primary: Alpöge, Bhargava, Ho and Shnidman, Rank stability in quadratic extensions and Hilbert's tenth problem
- Primary: Published Inventiones Mathematicae article
- Independent: Quanta, New Proofs Probe the Limits of Mathematical Truth
- Independent: Princeton publication record
- Independent: Peter Koymans publication list
- Independent: Carlo Pagano publication list
Correction and revision history
- 2026-07-22 — Added as two independent proof routes with the ring-of-integers domain stated precisely.
- 2026-07-22 — Corrected the review status: the Alpöge–Bhargava–Ho–Shnidman proof is published in Inventiones Mathematicae, and both authors list the independent Koymans–Pagano proof as forthcoming in JAMS.
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