The Genesis of a Theorem in the Galois Theory of <i>p</i>-Extensions of ℚ with Restricted Tame Ramification
Bibliographic record
Abstract
This article traces the genesis of a theorem that gives for the first time examples of the Galois group GS of the maximal p-extension of ℚ, unramified outside a finite set of primes not containing an odd p, that are of cohomlogical dimension 2 if the primes in S satisfy a certain linking condition. Because the ramification is tame the pro-p-group GS has all of its derived factors finite which is a strong finitenesss condition on GS. The paper starts with a question of Serre on one relator pro-p-groups and then a detour to discrete groups where the notion of strong freeness for a sequence of homogeneous Lie elements is given and a criterion for strong freeness is established. These notions are then carried over to pro-p-groups where the linking condtion on the primes of S is translated into a cohomological criterion for a pro-p-group to have cohomological dimension 2. An analysis is given of the work of Koch where he gives a weaker criterion for a pro-p-group to have have cohomological dimension 2. A connecttion is made with this work of Koch and that of the author which would have been sufficient to prove the fact that GS was of cohomological dimension 2 for certain sets S had it been applied to investigate whether the linking condition was true for certain sets S. It is not known if the cohomological dimension of GS is 2 if S does not satisfy this linking condition.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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 | 0.004 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.003 | 0.013 |
| Scholarly communication | 0.004 | 0.011 |
| Open science | 0.001 | 0.005 |
| Research integrity | 0.001 | 0.005 |
| 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".