No abstract

We study the p-rationality of real quadratic fields in terms of generalized Fibonacci numbers and their periods modulo positive integers.

Let k be an imaginary quadratic field and p an odd prime number such that the p-rank of the class group of k is one. Let S be a finite set of places of k distinct from p-adic places. We give sufficient conditions for the Galois group G S , of the maximal pro-p-extension of k which is unramified outside S, to be mild, hence of cohomological dimension 2.

Using half-integral weight modular forms we give a criterion for the existence of real quadratic p-rational fields. For p = 5 we prove the existence of infinitely many real quadratic p-rational fields.