Essential dimension of inseparable field extensions
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Abstract
Let [math] be a base field, [math] be a field containing [math] , and [math] be a field extension of degree [math] . The essential dimension [math] over [math] is a numerical invariant measuring “the complexity” of [math] . Of particular interest is\n¶\n<math display="block">\n<mrow>\n<mi>τ</mi>\n<mrow>\n<mo class="MathClass-open">(</mo>\n<mrow>\n<mi>n</mi>\n</mrow>\n<mo class="MathClass-close">)</mo>\n</mrow>\n<mo class="MathClass-rel">=</mo>\n<mo class="qopname"> max</mo>\n<mrow>\n<mo class="MathClass-open" fence="true" mathsize="1.19em">{</mo>\n<mrow>\n<mo class="qopname">ed</mo>\n<mrow>\n<mo class="MathClass-open">(</mo>\n<mrow>\n<mi>L</mi>\n<mo class="MathClass-bin">∕</mo>\n<mi>K</mi>\n</mrow>\n<mo class="MathClass-close">)</mo>\n</mrow>\n<mo class="MathClass-rel">∣</mo>\n<mtext/>\n<mi>L</mi>\n<mo class="MathClass-bin">∕</mo>\n<mi>K</mi>\n<mtext> is a separable extension of degree </mtext>\n<mi>n</mi>\n<mtext/>\n</mrow>\n<mo class="MathClass-close" fence="true" mathsize="1.19em">}</mo>\n</mrow>\n<mo class="MathClass-punc">,</mo>\n</mrow>\n</math>\n¶ also known as the essential dimension of the symmetric group [math] . The exact value of [math] is known only for [math] . In this paper we assume that [math] is a field of characteristic [math] and study the essential dimension of inseparable extensions [math] . Here the degree [math] is replaced by a pair [math] which accounts for the size of the separable and the purely inseparable parts of [math] , respectively, and [math] is replaced by\n¶\n<math display="block">\n<mrow>\n<mi>τ</mi>\n<mrow>\n<mo class="MathClass-open">(</mo>\n<mrow>\n<mi>n</mi>\n<mo class="MathClass-punc">,</mo>\n<mi>e</mi>\n</mrow>\n<mo class="MathClass-close">)</mo>\n</mrow>\n<mo class="MathClass-rel">=</mo>\n<mo class="qopname"> max</mo>\n<mrow>\n<mo class="MathClass-open" fence="true" mathsize="1.19em">{</mo>\n<mrow>\n<mo class="qopname">ed</mo>\n<mrow>\n<mo class="MathClass-open">(</mo>\n<mrow>\n<mi>L</mi>\n<mo class="MathClass-bin">∕</mo>\n<mi>K</mi>\n</mrow>\n<mo class="MathClass-close">)</mo>\n</mrow>\n<mo class="MathClass-rel">∣</mo>\n<mtext/>\n<mi>L</mi>\n<mo class="MathClass-bin">∕</mo>\n<mi>K</mi>\n<mtext> is a field extension of type </mtext>\n<mrow>\n<mo class="MathClass-open">(</mo>\n<mrow>\n<mi>n</mi>\n<mo class="MathClass-punc">,</mo>\n<mi>e</mi>\n</mrow>\n<mo class="MathClass-close">)</mo>\n</mrow>\n<mtext/>\n</mrow>\n<mo class="MathClass-close" fence="true" mathsize="1.19em">}</mo>\n</mrow>\n<mo class="MathClass-punc">.</mo>\n</mrow>\n</math>\n¶ The symmetric group [math] is replaced by a certain group scheme [math] over [math] . This group scheme is neither finite nor smooth; nevertheless, computing its essential dimension turns out to be easier than computing the essential dimension of [math] . Our main result is a simple formula for [math] .
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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) | 0.001 | 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 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.046 | 0.002 |
Machine scores (provisional)
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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 it