最简四次域之间的等式：完整分类
Equalities among simplest quartic fields: a complete classification
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中文总结 AI 辅助
该研究完整分类了最简四次域Kₙ的重合情况，确定仅存在三对不同正整数对应的Kₘ与Kₙ相等，通过结合Thue方程对应关系与有理逼近估计完成证明，扩展了相关分类结果。
中文摘要 AI 辅助
对于正整数n，令fₙ(X)=X⁴−nX³−6X²+nX+1，且令Kₙ=ℚ(ρₙ)，其中ρₙ是fₙ的根。我们确定这些域之间的所有重合：对于不同的正整数m、n，Kₘ=Kₙ当且仅当{m,n}属于{{1,103},{2,22},{4,956}}。因此，此前已知的三个等式是仅有的等式。这将Hoshi的有限范围分类扩展到所有正整数参数，尤其包含了Pincus和Washington的唯一性结果。证明结合了Hoshi关于相等最简四次域与四次Thue方程本原解的对应关系，以及Lettl–Pethő–Voutier对其两个实根有理逼近的估计；一个高斯整数恒等式给出了所得有理逼近分母的下界，连分数信息和逼近估计随后排除了所有超过1000的参数。
英文摘要
For a positive integer $n$, let $f_n(X)=X^4-nX^3-6X^2+nX+1$ and let $K_n=\mathbb{Q}(ρ_n)$, where $ρ_n$ is a root of $f_n$. We determine all coincidences among these fields: for distinct positive integers $m,n$, $K_m=K_n \Longleftrightarrow \{m,n\}\in\{\{1,103\},\{2,22\},\{4,956\}\}$. Thus the three previously known equalities are the only ones. This extends Hoshi's finite-range classification to all positive integral parameters and, in particular, subsumes the uniqueness results of Pincus and Washington. The proof combines Hoshi's correspondence between equal simplest quartic fields and primitive solutions of a quartic Thue equation with estimates of Lettl--Pethő--Voutier for rational approximations to two of its real roots. A Gaussian-integer identity yields a lower bound for the denominator of the resulting rational approximation; the continued-fraction information and the approximation estimates then exclude every parameter exceeding $1000$.
