The p-rationality of a totally real abelian number field can be checked from the values $L(2-p,\chi )$ of Dirichlet L-functions, for all non-principal even Dirichlet characters associated to the field. Using this criterion and the properties of the generalized Bernoulli numbers, we study the p-rationality of $\mathbb{Q}{({\zeta }_{2l+1})}^{+}$, the maximal real subfield of $\mathbb{Q}({\zeta }_{2l+1})$, for Sophie Germain primes l and odd primes p that are primitive roots modulo l. We prove that $\mathbb{Q}{({\zeta }_{2l+1})}^{+}$ is p-rational for such pairs if $p<4l$. We also prove that the Siegel's heuristics on the equidistribution of the residues of Bernoulli numbers modulo p imply that $\mathbb{Q}{({\zeta }_{2l+1})}^{+}$ is p-rational for all but finitely many p that are primitive roots modulo l.

可以从值中检查完全实数阿贝尔数域的p 有理性$升(2-p,\chi )$狄利克雷L函数，适用于与该域相关的所有非主要甚至狄利克雷字符。使用这个标准和广义伯努利数的性质，我们研究了p 的合理性$\mathbb{问}{({\zeta }_{2升+1})}^{+}$, 的最大实子域 $\mathbb{问}({\zeta }_{2升+1})$, 对于 Sophie Germain 素数l和奇素数p是原始根模l。我们证明$\mathbb{问}{({\zeta }_{2升+1})}^{+}$对于这些对是p -有理的，如果$p<4升$. 我们还证明了关于以p为模的伯努利数的余数的等分布的 Siegel 启发式意味着$\mathbb{问}{({\zeta }_{2升+1})}^{+}$对于除有限多个p是原始根模l之外的所有p是p有理的。