AI 中文总结
研究伽罗瓦扩张中素理想竞赛问题，通过引入代数参数给出偏差对数密度位于特定区间的条件特征，构造出偏差源于\(2p\)次根数量差异的伽罗瓦扩张，给出最小群阶及广义构造，还使一估计无条件成立。
AI 中文摘要
设\(L/K\)为数域的伽罗瓦扩张，\(C_1,C_2\)为\(\mathrm{Gal}(L/K)\)的共轭类。通过引入与\(C_1\)和\(C_2\)相关的代数参数，我们给出了切比雪夫偏差对数密度\(\delta_{L/K}(C_1,C_2)\)位于区间\((1/2,1)\)（即素理想竞赛有偏差）的条件特征。与文献中现有例子不同，我们用此准则构造伽罗瓦扩张，偏差完全源于奇素数\(p\)的\(2p\)次根数量差异。证明了\(p = 3\)时此现象所需最小伽罗瓦群阶为\(96\)，\(p = 5\)时为\(320\)，且给出了所有奇素数的广义构造。还证明了青木和小山在深度黎曼假设下建立的一个估计在某些情况下无条件成立。
英文摘要
Let $L/K$ be a Galois extension of number fields, and let $C_1, C_2$ be conjugacy classes of $\mathrm{Gal}(L/K)$. By introducing algebraic parameters related to $C_1$ and $C_2$, we provide a conditional characterization for Chebyshev's bias logarithmic density $δ_{L/K}(C_1,C_2)$ to lie in the interval $(1/2, 1)$, meaning that the prime ideal race is biased. Unlike existing examples in the literature, where the bias comes from a difference in the number of square roots, the order of vanishing of Artin $L$-functions at $s=1/2$, or a combination of both, we use this criterion to construct Galois extensions where neither of these aspects plays a role. In our constructions, the bias arises entirely from a difference in the number of $2p$-th roots for an odd prime $p$. We prove that the minimal Galois group order required for this phenomenon is $96$ for $p=3$ and $320$ for $p=5$, where in both cases the group structure is a direct product of two generalized quaternion groups. Furthermore, we provide generalized constructions for all odd primes. Finally, these same algebraic parameters enable us to prove that an estimate established by Aoki and Koyama under the Deep Riemann Hypothesis holds unconditionally in certain cases.
