Galois module structure of (ℓ<sup> <i>n</i> </sup>)th classes of fields
Bibliographic record
Abstract
In this paper, we use the Merkurjev–Suslin theorem to determine the structure of arithmetically significant Galois modules that arise from Kummer theory. Let K be a field of characteristic different from a prime ℓ, n be a positive integer, and suppose that K contains the (ℓn)th roots of unity. Let L be the maximal ℓn-elementary abelian extension of K, and set G=Gal(L| K). We consider the G-module J≔L×/ℓn and denote its socle series by Jm. We provide a precise condition, in terms of a map to H3(G, ℤ/ℓn), determining which submodules of Jm−1 embed in cyclic modules generated by elements of Jm; therefore, this map provides an explicit description of Jm and Jm/Jm−1. The description of Jm/Jm−1 is a new non-trivial variant of the classical Hilbert's Theorem 90. The main theorem generalizes a theorem of Adem, Gao, Karaguezian and Mináč that deals with the case m=ℓn=2, and also ties in with current trends in minimalistic birational anabelian geometry over essentially arbitrary fields.
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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 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| 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) | 0.008 | 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".