Galois groups over nonrigid fields
Bibliographic record
Abstract
. Let F be a field with char F 6= 2. We show that F is a nonrigid field if and only if certain small 2-groups occur as Galois groups over F . These results provide new "automatic realizability" results for Galois groups over F . The groups we consider demonstrate the inequality of two particular metabelian 2-extensions of F which are unequal precisely when F is a nonrigid field. Using known results on connections between rigidity and existence of certain valuations, we obtain Galois-theoretic criteria for the existence of these valuations. 1. Introduction Let F be a field with char F 6= 2. The goal of this paper is to identify basic Galois groups that must occur if F is not a rigid field. If j F= F 2 j 8 then it is known that F is not rigid if and only if a certain group (often denoted DC) of order 16 occurs as a Galois group over F ([MS]). It has also been shown ([LS], simplifying an earlier proof in [AGKM]) that F is rigid if and only if F (3) = F f3g (all notatio...
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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 | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.016 | 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".