On the role of higher roots in prime ideal races

Abstract

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 $\delta_{L/K}(C_1,C_2)$ 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.