Primitive normal pairs with prescribed traces over finite fields
Bibliographic record
Abstract
<p>Let <span class="math inline">\(q\)</span> be a positive integral power of some prime <span class="math inline">\(p\)</span> and <span class="math inline">\(\mathbb{F}_{q^m}\)</span> be a finite field with <span class="math inline">\(q^m\)</span> elements for some <span class="math inline">\(m \in \mathbb{N}\)</span>. Here we establish a sufficient condition for the existence of primitive normal pairs of the type <span class="math inline">\((\epsilon, f(\epsilon))\)</span> in <span class="math inline">\(\mathbb{F}_{q^m}\)</span> over <span class="math inline">\(\mathbb{F}_{q}\)</span> with two prescribed traces, <span class="math inline">\(\text{Tr}_{{\mathbb{F}_{q^m}}/{\mathbb{F}_q}}(\epsilon)=a\)</span> and <span class="math inline">\(\text{Tr}_{{\mathbb{F}_{q^m}}/{\mathbb{F}_q}}(f(\epsilon))=b\)</span>, where <span class="math inline">\(f(x) \in \mathbb{F}_{q^m}(x)\)</span> is a rational function with some restrictions and <span class="math inline">\(a, b \in \mathbb{F}^*_q\)</span>. Furthermore, for <span class="math inline">\(q=5^k\)</span>, <span class="math inline">\(m \geq 9\)</span> and rational functions with degree sum 4, we explicitly find at most 13 fields in which the desired pair may not exist.</p>
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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 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 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 | 0.000 |
| 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".