Existence of normal elements with prescribed norms
Bibliographic record
Abstract
For each positive integer n, let Fqn be the unique n-degree extension of the finite field Fq with q elements, where q is a prime power. It is known that for arbitrary q and n, there exist elements β∈Fqn such that the Galois conjugates β,βq,…,βqn−1 form a basis for Fqn as an Fq-vector space. These elements are called normal and they work as additive generators of finite fields. On the other hand, the multiplicative group F∗qn is cyclic and any generator of this group is a primitive element. Many past works have dealt with the existence of primitive and normal elements with specified properties, including the existence of primitive elements whose traces over intermediate extensions are prescribed. Inspired by the latter, we explore the existence of normal elements whose norms over intermediate extensions are prescribed. We combine combinatorial and number-theoretic ideas and obtain both asymptotic and concrete results. In particular, we completely solve the problem in the case where only one intermediate extension is considered.
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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 | 0.002 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.002 | 0.001 |
| Research integrity | 0.000 | 0.000 |
| 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".