8-rank Of Narrow Class Groups For Some Real Quadratic Number Fields

It is one of important subjects in algebraic number theory to study the class number of a number field. In a decomposition of the 2-Sylow subgroup of the class group of a quadratic number field into cyclic subgroup summands, counting the number of summands of order at most 8, namely the 8-rank of class group, becomes significant.This paper mainly investigates the 8-ranks of real quadratic fields (?), where d =δpq , p≡q≡1mod 4and d =δp₁p₂p₃, p_i≡1mod 4, i= 1,2,3,δ∈{1,2}, are squarefree positive integers, gives the necessary and sufficient conditions of 8-ranks of their narrow class groups equal to 1 or 2, and determines whether the norms of their fundamental units are 1 or -1 in certain cases. This paper uses genus theory, Rédei matrix, Gauss theorem, and Legendre theorem to get these results. They are in terms of congruence relations modulo prime, Legendre symbol and quartic residue symbol and so on. They are very useful for numerical computations.