有限域上的Krasner商：同构阈值、特征与普查
Finite-field Krasner quotients: isomorphism thresholds, characteristics, and censuses
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中文总结 AI 辅助
该研究确定有限域Krasner商的特征，给出同构阈值、经验稳定界等计算结果，发现阶7的有限域商仅15种，为相关猜想提供数据支持。
中文摘要 AI 辅助
我们研究由有限域产生的Krasner商超域 $F_q/G_r$，其中 $G_r\le F_q^\times$ 的指数为 $r$。基于Baker-Jin的结构定理，我们确定所有足够大的此类商的特征与C-特征：它们仅依赖于 $r$ 的奇偶性，且当 $r$ 为偶数时，还依赖于 $q$ 模 $2r$ 的剩余类。特别地，对于偶数 $r$，Baker-Jin的两类稳定类由特征 $2$ 与 $3$ 区分，而C-特征始终为 $1$。我们补充了一个计算实验室：Baker-Jin大-$q$同构的精确Weil阈值、经验最小稳定界 $N_r^{\mathrm{emp}}$、阶 $n\le7$ 的有限域商完整超域图集，以及与Ameri-Eyvazi-Hošková-Mayerová（阶 $\le6$）和Massouros-Massouros（阶 $7$）的枚举结果的对比。在其他发现中，阶 $7$ 的超域共 $277$ 种，其中恰好 $15$ 种同构类型作为有限域商出现——这是支持Baker-Jin商稀有性猜想的具体数据点。所有算法与表格均包含在开源包中，可供独立验证及作为arXiv辅助材料。
英文摘要
We study Krasner quotient hyperfields arising from finite fields, $F_q/G_r$, where $G_r\le F_q^\times$ has index $r$. Building on the structure theorem of Baker--Jin, we determine the characteristic and C-characteristic of all sufficiently large such quotients: they depend only on the parity of $r$ and, when $r$ is even, on the residue class of $q$ modulo $2r$. In particular, the two Baker--Jin stable classes for even $r$ are separated by characteristic $2$ versus $3$, while the C-characteristic is always $1$. We complement this structural result with a computational laboratory: sharp Weil thresholds for Baker--Jin large-$q$ isomorphism, empirical minimal stabilization bounds $N_r^{\mathrm{emp}}$, complete finite-field quotient atlases for hyperfield orders $n\le 7$, and comparisons with the enumerations of Ameri--Eyvazi--Hošková-Mayerová (orders $\le 6$) and Massouros--Massouros (order $7$). Among other findings, exactly $15$ isomorphism types of order $7$ arise as finite-field quotients, out of $277$ hyperfields of that order -- a concrete data point toward the Baker--Jin rarity conjecture for quotients. All algorithms and tables are available in an open-source package suitable for independent verification and arXiv ancillary material.
