Finite index theorems for iterated Galois groups of unicritical polynomials
Bibliographic record
Abstract
Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the function field of a smooth irreducible curve defined over <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Q overbar"> <mml:semantics> <mml:mover> <mml:mi>Q</mml:mi> <mml:mo accent="false"> ¯ </mml:mo> </mml:mover> <mml:annotation encoding="application/x-tex">\overline {Q}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f element-of upper K left-bracket x right-bracket"> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo> ∈ </mml:mo> <mml:mi>K</mml:mi> <mml:mo stretchy="false">[</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">f\in K[x]</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be of the form <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f left-parenthesis x right-parenthesis equals x Superscript q Baseline plus c"> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:msup> <mml:mi>x</mml:mi> <mml:mi>q</mml:mi> </mml:msup> <mml:mo>+</mml:mo> <mml:mi>c</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">f(x)=x^q+c</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q equals p Superscript r Baseline comma r greater-than-or-equal-to 1 comma"> <mml:semantics> <mml:mrow> <mml:mi>q</mml:mi> <mml:mo>=</mml:mo> <mml:msup> <mml:mi>p</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>r</mml:mi> </mml:mrow> </mml:msup> <mml:mo>,</mml:mo> <mml:mi>r</mml:mi> <mml:mo> ≥ </mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">q = p^{r}, r \ge 1,</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a power of the prime number <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , and let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="beta element-of upper K overbar"> <mml:semantics> <mml:mrow> <mml:mi> β </mml:mi> <mml:mo> ∈ </mml:mo> <mml:mover> <mml:mi>K</mml:mi> <mml:mo accent="false"> ¯ </mml:mo> </mml:mover> </mml:mrow> <mml:annotation encoding="application/x-tex">\beta \in \overline {K}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . For all <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n element-of double-struck upper N union StartSet normal infinity EndSet"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo> ∈ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">N</mml:mi> </mml:mrow> <mml:mo> ∪ </mml:mo> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mi mathvariant="normal"> ∞ </mml:mi> <mml:mo fence="false" stretchy="false">}</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">n\in \mathbb {N}\cup \{\infty \}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , the Galois groups <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G Subscript n Baseline left-parenthesis beta right-parenthesis equals normal upper G normal a normal l left-parenthesis upper K left-parenthesis f Superscript negative n Baseline left-parenthesis beta right-parenthesis right-parenthesis slash upper K left-parenthesis beta right-parenthesis right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>G</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi> β </mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">G</mml:mi> <mml:mi mathvariant="normal">a</mml:mi> <mml:mi mathvariant="normal">l</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>K</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mi>f</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo> − </mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi> β </mml:mi> <mml:mo stretchy="false">)</mml:mo>
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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 | 0.002 | 0.002 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.003 | 0.004 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.004 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.002 | 0.000 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.001 | 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".