Bibliographic record
Abstract
Abstract. Let q be an algebraic integer of degree d ≥ 2. Consider the rank of the multiplicative subgroup of ℂ* generated by the conjugates of q . We say q is of full rank if either the rank is d − 1 and q has norm ±1, or the rank is d . In this paper we study some properties of ℤ [ q ] where q is an algebraic integer of full rank. The special cases of when q is a Pisot number and when q is a Pisot-cyclotomic number are also studied. There are four main results. (1) If q is an algebraic integer of full rank and n is a fixed positive integer, then there are only finitely many m such that disc `ℤ [ q m ]´ = disc `ℤ[ q n ] ´. (2) If q and r are algebraic integers of degree d of full rank and ℤ [ q n ] = ℤ[ r n ] for infinitely many n , then either q = ω r ′ or q = Norm( r ) 2/ d ω/ r ′ , where r ′ is some conjugate of r and ω is some root of unity. (3) Let r be an algebraic integer of degree at most 3. Then there are at most 40 Pisot numbers q such that ℤ [ q ] = ℤ[ r ] . (4) There are only finitely many Pisot-cyclotomic numbers of any fixed order.
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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 | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 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) | 0.000 | 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".