Cubic fields, a congruential criterion for Scholz's theorem and new real quadratic fields with 3-rank equal to 4
Let k be a quadratic field of discriminant d and let \(H_ 3(d)\) be the subgroup of elements having order three of its ideal class group H(d). Let k be real; k' is defined as its associated imaginary quadratic field of discriminant \(d'=-3d/(3,d)\) 2. We denote by s and r the rank of the group \(H_ 3(d)\) and \(H_ 3(d')\), respectively. \textit{A. Scholz} [J. Reine Angew. Math. 166, 201-203 (1932; Zbl 0004.05104)] proves that: \(s\leq r\leq s+1.\) The authors compute s from the subgroup \(H_ 3(d')\) of order-three elements of H(d'). They give also the first known examples of quadratic fields with \(r=s=4\). He gives finally 40 non isomorphic totally real cubic fields K with discriminant \(d_ 0=314582172161\).
scientific article; zbMATH DE number 4045875
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