FIELDS OF DEFINITION FOR DIVISION ALGEBRAS
Bibliographic record
Abstract
Let A be a finite-dimensional division algebra containing a base field k in its center F. A is defined over a subfield F0 if there exists an F0-algebra A0 such that A = A 0 ⊗ F 0 F . The following are shown. (i) In many cases A can be defined over a rational extension of k. (ii) If A has odd degree n ⩾ 5, then A is defined over a field F0 of transcendence degree ⩽ 1/2(n−1)(n−2) over k. (iii) If A is a Z/m × Z/2-crossed product for some m ⩾ 2 (and in particular, if A is any algebra of degree 4) then A is Brauer equivalent to a tensor product of two symbol algebras. Consequently, Mm(A) can be defined over a field F0 such that trdegk(F0) ⩽ 4. (iv) If A has degree 4 then the trace form of A can be defined over a field F0 of transcendence degree ⩽ 4. (In (i), (iii) and (iv) it is assumed that the center of A contains certain roots of unity.)
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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 | 0.004 | 0.005 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.003 | 0.003 |
| Science and technology studies | 0.003 | 0.005 |
| Scholarly communication | 0.006 | 0.012 |
| Open science | 0.002 | 0.004 |
| Research integrity | 0.002 | 0.005 |
| Insufficient payload (model declined to judge) | 0.015 | 0.006 |
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".