On p-rationality of [formula omitted] for Sophie Germain primes l

Abstract

The p-rationality of a totally real abelian number field can be checked from the values L(2−p,χ) 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 Q(ζ2l+1)+, the maximal real subfield of Q(ζ2l+1), for Sophie Germain primes l and odd primes p that are primitive roots modulo l. We prove that Q(ζ2l+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 Q(ζ2l+1)+ is p-rational for all but finitely many p that are primitive roots modulo l.