Triple
T9297066
| Position | Surface form | Disambiguated ID | Type / Status |
|---|---|---|---|
| Subject | Chebotarev density theorem |
E223663
|
entity |
| Predicate | strengthens |
P3261
|
FINISHED |
| Object |
Frobenius density theorem
The Frobenius density theorem is a result in algebraic number theory that describes how often prime ideals in a number field have a given Frobenius conjugacy class, providing a precursor to the more general Chebotarev density theorem.
|
E223663
|
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: Frobenius density theorem | Statement: [Chebotarev density theorem, strengthens, Frobenius density theorem]
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Frobenius density theorem Context triple: [Chebotarev density theorem, strengthens, Frobenius density theorem]
-
A.
Chebotarev density theorem
The Chebotarev density theorem is a fundamental result in algebraic number theory that generalizes the prime number theorem to describe how often primes in a number field have a given Frobenius conjugacy class in its Galois group.
-
B.
Artin’s conjecture on L-functions
Artin’s conjecture on L-functions is a major unproven hypothesis in number theory asserting that nontrivial Artin L-functions associated to Galois representations are entire, with deep implications for the distribution of primes and the structure of number fields.
-
C.
Sato–Tate distribution (for families of elliptic curves)
The Sato–Tate distribution (for families of elliptic curves) is a probabilistic law describing how the normalized Frobenius traces (or equivalently, the angles in the Hasse bound) of elliptic curves are distributed, typically following a specific sine-squared measure on the interval [0, π].
-
D.
Siegel’s theorem on zeros of L-functions
Siegel’s theorem on zeros of L-functions is a result in analytic number theory that gives strong bounds on how close nontrivial zeros of Dirichlet L-functions can approach 1, with deep implications for the distribution of primes in arithmetic progressions.
-
E.
Hasse–Arf theorem
The Hasse–Arf theorem is a fundamental result in algebraic number theory that precisely characterizes the jumps in the ramification filtration of abelian extensions of local fields, showing they occur at integer values.
- 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: Frobenius density theorem Triple: [Chebotarev density theorem, strengthens, Frobenius density theorem]
Generated description
The Frobenius density theorem is a result in algebraic number theory that describes how often prime ideals in a number field have a given Frobenius conjugacy class, providing a precursor to the more general Chebotarev density theorem.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Frobenius density theorem Target entity description: The Frobenius density theorem is a result in algebraic number theory that describes how often prime ideals in a number field have a given Frobenius conjugacy class, providing a precursor to the more general Chebotarev density theorem.
-
A.
Chebotarev density theorem
chosen
The Chebotarev density theorem is a fundamental result in algebraic number theory that generalizes the prime number theorem to describe how often primes in a number field have a given Frobenius conjugacy class in its Galois group.
-
B.
Artin’s conjecture on L-functions
Artin’s conjecture on L-functions is a major unproven hypothesis in number theory asserting that nontrivial Artin L-functions associated to Galois representations are entire, with deep implications for the distribution of primes and the structure of number fields.
-
C.
Sato–Tate distribution (for families of elliptic curves)
The Sato–Tate distribution (for families of elliptic curves) is a probabilistic law describing how the normalized Frobenius traces (or equivalently, the angles in the Hasse bound) of elliptic curves are distributed, typically following a specific sine-squared measure on the interval [0, π].
-
D.
Siegel’s theorem on zeros of L-functions
Siegel’s theorem on zeros of L-functions is a result in analytic number theory that gives strong bounds on how close nontrivial zeros of Dirichlet L-functions can approach 1, with deep implications for the distribution of primes in arithmetic progressions.
-
E.
Hasse–Arf theorem
The Hasse–Arf theorem is a fundamental result in algebraic number theory that precisely characterizes the jumps in the ramification filtration of abelian extensions of local fields, showing they occur at integer values.
- F. None of above.
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_69ca8423edb08190bc0c91287a484768 |
completed | March 30, 2026, 2:09 p.m. |
| NER | Named-entity recognition | batch_69cd089c9c588190967404c9eb938dfb |
completed | April 1, 2026, 11:59 a.m. |
| NED1 | Entity disambiguation (via context triple) | batch_69d0b251c4148190a94fafdc23a601d6 |
completed | April 4, 2026, 6:40 a.m. |
| NEDg | Description generation | batch_69d0b65ea4548190b445563ac695b008 |
completed | April 4, 2026, 6:57 a.m. |
| NED2 | Entity disambiguation (via description) | batch_69d0b6f504a48190878db828312e8a97 |
completed | April 4, 2026, 7 a.m. |
Created at: March 30, 2026, 7:36 p.m.