Jacobi sums and new families of irreducible polynomials of Gaussian periods
Bibliographic record
Abstract
Let m > 2 m> 2 , ζ m \zeta _m an m m -th primitive root of 1, q ≡ 1 q\equiv 1 mod 2 m 2m a prime number, s = s q s=s_{q} a primitive root modulo q q and f = f q = ( q − 1 ) / m f=f_{q}=(q-1)/m . We study the Jacobi sums J a , b = − ∑ k = 2 q − 1 ζ m a ind s ( k ) + b ind s ( 1 − k ) J_{a,b}=-\sum _{k=2}^{q-1}\zeta _m ^{\, a\, \text {ind}_{s}(k)+b\, \text {ind}_{s}(1-k)} , 0 ≤ a , b ≤ m − 1 0\leq a, b\leq m-1 , where ind s ( k ) \text {ind}_{s}(k) is the least nonnegative integer such that s ind s ( k ) ≡ k s^{\, \text {ind}_{s}(k)}\equiv k mod q q . We exhibit a set of properties that characterize these sums, some congruences they satisfy, and a MAPLE program to calculate them. Then we use those results to show how one can construct families P q ( x ) P_{q}(x) , q ∈ P q\in \mathcal {P} , of irreducible polynomials of Gaussian periods,
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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 | 0.001 | 0.005 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.003 | 0.002 |
| Science and technology studies | 0.002 | 0.003 |
| Scholarly communication | 0.003 | 0.004 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.013 | 0.003 |
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".