Principal well-rounded ideals of real quadratic fields
Bibliographic record
Abstract
It has been well known since Gauss that the principality of an ideal in a real quadratic field [Formula: see text] is equivalent to the solvability of a certain generalized Pell equations. In this paper, we combine this classical result with Srinivasan’s conditions for the existence of well-rounded ideals in [Formula: see text] to obtain necessary and sufficient criteria for a real quadratic field to have principal well-rounded (PWR) ideals. Using these criteria, we prove that there are infinitely many real quadratic fields that have PWR ideals. Moreover, these ideals are pairwise non-similar. We then construct new algorithms that produce these PWR ideals, especially when the field discriminant is large. Our algorithms run in sub-exponential time theoretically; however, they are very fast in practice by employing some commonly used probabilistic algorithms for testing squarefreeness. Finally, we briefly consider criteria for the existence of prime PWR ideals and show that there are infinitely many real quadratic fields that have prime PWR ideals.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.003 | 0.011 |
| Meta-epidemiology (narrow) | 0.000 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.003 | 0.006 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.000 | 0.002 |
| Insufficient payload (model declined to judge) | 0.006 | 0.001 |
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 source (direct Gemma or distilled Codex), 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".