Vol. 108, No. 1, 1983

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The quadratic number fields with cyclic 2-classgroups

Richard Patrick Morton

Vol. 108 (1983), No. 1, 165–175

Many authors have considered the divisibility of the restricted class number h+(d) of the quadratic field Ω = Q(√d-) by 4 and 8, in the case that the discriminant d of Ω has exactly two prime factors. For such discriminants the restricted classgroup 𝒞 of Ω has a nontrivial cyclic 2-Sylow subgroup, and conditions on d can be given for the existence of classes in 𝒞 of orders 4 and 8. The first such results are due to Rédei.

In this paper we give criteria for the divisibility of h+(d) by 8 which are phrased in terms of the splitting of one of the prime factors p of d in a normal extension of Q depending only on d∕p = d0. This simplifies and unifies the criteria for 8h+(d) existing in the literature, which depend mainly in the representation of the prime p by certain quadratic forms, or on the quadratic character of solutions to ternary quadratic equations.

Mathematical Subject Classification
Primary: 12A25, 12A25
Secondary: 12A50
Received: 6 November 1981
Revised: 28 April 1982
Published: 1 September 1983
Richard Patrick Morton