COMMON DIVISORS OF VALUES OF POLYNOMIALS AND COMMON FACTORS OF INDICES IN A NUMBER FIELD
Bibliographic record
Abstract
Let K be a number field of degree n over ℚ, Â be the set of integers of K that are primitive over ℚ and let I(K) be its index. The prime factors of I(K) are called common factors of indices or inessential discriminant divisors. We show that these primes divide another index i(K) previously defined by Gunji and McQuillan as i(K) = lcm θ∈Â i(θ), where i(θ) = gcd x∈ℤ F θ (x) and F θ (x) is the characteristic polynomial of θ over ℚ. It is shown that there exists θ ∈ Â such that i(K) = i(θ) and an algorithm is given for the computation of such an integer. For any prime p|i(K), an integer ρ K (p) defined as the number of [Formula: see text] such that p|i(θ) is investigated. It is shown that this integer determines in some cases the splitting type of p in K. Some open questions related to I(K), i(K) and ρ K (p) are stated.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.003 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".