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Construction of the real dihedral number fields of degree 2p. Applications

1. Introduction. We use class field theory to construct the real dihedral fields. This construction is reduced to that of primitive characters on the ring class groups of real quadratic fields which are then used to compute the relative class numbers of dihedral CM-fields of degree 4p, p any odd prime. We first fix some notation. Let K be a normal real number field (considered as a subfield of the field of complex numbers) of degree 2n with Galois group D 2n = a, b : a