Shirshov's theorem and division rings that are left algebraic over a\n subfield
Bibliographic record
Abstract
Let D be a division ring. We say that D is left algebraic over a (not\nnecessarily central) subfield K of D if every x in D satisfies a polynomial\nequation x^n + a_{n-1}x^{n-1}+...+a_0=0 with a_0,...,a_{n-1} in K. We show that\nif D is a division ring that is left algebraic over a subfield K of bounded\ndegree d then D is at most d^2-dimensional over its center. This generalizes a\nresult of Kaplansky. For the proof we give a new version of the combinatorial\ntheorem of Shirshov that sufficiently long words over a finite alphabet contain\neither a q-decomposable subword or a high power of a non-trivial subword. We\nshow that if the word does not contain high powers then the factors in the\nq-decomposition may be chosen to be of almost the same length. We conclude by\ngiving a list of problems for algebras that are left algebraic over a\ncommutative subring.\n
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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 | 0.001 |
| Open science | 0.002 | 0.003 |
| Research integrity | 0.000 | 0.001 |
| 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".