Triple

T4597225
Position Surface form Disambiguated ID Type / Status
Subject Kronecker–Weber theorem E100232 entity
Predicate standardReference P33736 FINISHED
Object Neukirch: Algebraic Number Theory
"Neukirch: Algebraic Number Theory" is a widely respected graduate-level textbook that provides a rigorous, modern introduction to algebraic number theory, including class field theory and foundational results such as the Kronecker–Weber theorem.
E459564 NE FINISHED

How this triple was built (4 steps)

Every LLM step that produced this triple, in pipeline order — named-entity classification, the disambiguation choices (the exact options shown, with the pick highlighted), and the generated description. The batch + timestamp of each is in the Provenance table below.

NER Named-entity recognition gpt-5-mini
Instruction
Given a phrase, classify it is english named entity (e.g., persons, organizations, works of art) in Latin script, or not (e.g., literals, dates, URLs, verbose phrases). For disambiguation, the statement where the phrase occurs as object is also given. Please return a JSON object with `phrase` (string, the phrase being analyzed) and `is_ne` (boolean, indicating whether the phrase is a Named Entity).
Input
Phrase: Neukirch: Algebraic Number Theory | Statement: [Kronecker–Weber theorem, standardReference, Neukirch: Algebraic Number Theory]
NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07
Target entity: Neukirch: Algebraic Number Theory
Context triple: [Kronecker–Weber theorem, standardReference, Neukirch: Algebraic Number Theory]
  • A. A Course in Arithmetic
    A Course in Arithmetic is a classic introductory text in number theory by Jean-Pierre Serre, renowned for its concise and elegant treatment of fundamental arithmetic and algebraic concepts.
  • B. Die Theorie der algebraischen Zahlkörper
    "Die Theorie der algebraischen Zahlkörper" is a foundational mathematical monograph on algebraic number fields, authored by David Hilbert and published as part of his influential Zahlbericht.
  • C. Algebraic Groups and Class Fields
    "Algebraic Groups and Class Fields" is a influential mathematical monograph that develops the deep connections between algebraic group theory and class field theory within number theory and arithmetic geometry.
  • D. Hilbert’s twelfth problem
    Hilbert’s twelfth problem is one of David Hilbert’s famous list of 23 problems, asking for a general explicit class field theory that would generate all abelian extensions of a given number field using special values of analytic functions.
  • E. An Introduction to the Theory of Numbers
    An Introduction to the Theory of Numbers is a classic textbook in number theory, co-authored by G. H. Hardy, that systematically develops fundamental concepts such as divisibility, prime numbers, Diophantine equations, and quadratic forms.
  • F. None of above. chosen
  • G. Unsure - the case is ambiguous/there is not enough information to decide.
NEDg Description generation gpt-5.1
Instruction
Generate a one-sentence description of the target entity. 
You are given a context triple in the form (subject, predicate, object), where the object is the target entity. 
# Instructions
Use the triple to infer relevant information about the entity. Describe the entity based on what is most defining, well-known. 
Avoid repeating the information from the triple, unless really essential.
# Response Format
Return only the sentence: "Description: [one-sentence description of the target entity]"
Input
Entity: Neukirch: Algebraic Number Theory
Triple: [Kronecker–Weber theorem, standardReference, Neukirch: Algebraic Number Theory]
Generated description
"Neukirch: Algebraic Number Theory" is a widely respected graduate-level textbook that provides a rigorous, modern introduction to algebraic number theory, including class field theory and foundational results such as the Kronecker–Weber theorem.
NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07
Target entity: Neukirch: Algebraic Number Theory
Target entity description: "Neukirch: Algebraic Number Theory" is a widely respected graduate-level textbook that provides a rigorous, modern introduction to algebraic number theory, including class field theory and foundational results such as the Kronecker–Weber theorem.
  • A. A Course in Arithmetic
    A Course in Arithmetic is a classic introductory text in number theory by Jean-Pierre Serre, renowned for its concise and elegant treatment of fundamental arithmetic and algebraic concepts.
  • B. Die Theorie der algebraischen Zahlkörper
    "Die Theorie der algebraischen Zahlkörper" is a foundational mathematical monograph on algebraic number fields, authored by David Hilbert and published as part of his influential Zahlbericht.
  • C. Algebraic Groups and Class Fields
    "Algebraic Groups and Class Fields" is a influential mathematical monograph that develops the deep connections between algebraic group theory and class field theory within number theory and arithmetic geometry.
  • D. Hilbert’s twelfth problem
    Hilbert’s twelfth problem is one of David Hilbert’s famous list of 23 problems, asking for a general explicit class field theory that would generate all abelian extensions of a given number field using special values of analytic functions.
  • E. An Introduction to the Theory of Numbers
    An Introduction to the Theory of Numbers is a classic textbook in number theory, co-authored by G. H. Hardy, that systematically develops fundamental concepts such as divisibility, prime numbers, Diophantine equations, and quadratic forms.
  • F. None of above. chosen

Provenance (5 batches)

The batch behind each pipeline step, in order, with when it ran. Timestamps are batch-level — stages were processed in waves, so the object chain (NER → NED1 → NEDg → NED2) reads in order, but predicate / elicitation batches can sit in a different wave.

Step Stage Batch ID Status When
creating Elicitation batch_69bd43cbc014819098b45f435908f88a completed March 20, 2026, 12:55 p.m.
NER Named-entity recognition batch_69bd594055dc8190a50f1b4be2be1ba0 completed March 20, 2026, 2:27 p.m.
NED1 Entity disambiguation (via context triple) batch_69bdfa4a99c88190b7332fd2e1799b3a completed March 21, 2026, 1:54 a.m.
NEDg Description generation batch_69bdfb83b5d08190b2d8502e763a0841 completed March 21, 2026, 1:59 a.m.
NED2 Entity disambiguation (via description) batch_69bdfc0e456c81908efa3858d981ccc0 completed March 21, 2026, 2:01 a.m.
Created at: March 20, 2026, 1:11 p.m.