On Rédei matrices with minimal rank (Q2710508)

Let \(k\) be a quadratic number field with discriminant \(d\). Let \(r_4^+(k)\) denote the narrow 4-class rank of \(k\). Let \(d= p_1^*\cdots p_s^*\cdot p_{s+1}^*\cdots p_{s+t}^*\) be the decomposition to a product of prime discriminants with \(p_1^*\cdots p_s^*\) positive and \(p_{s+1}^*\cdots p_{s+t}^*\) negative. \textit{L. Rédei} and \textit{H. Reichardt} [J. Reine Angew. Math. 170, 69-74 (1933; Zbl 0007.39602)]and \textit{L. Rédei} [ibid. 171, 55-60 (1934; Zbl 0009.05101)] gave an inequality \(r_4^+(k)\leq s+ [\frac{t-1}{2}]\), and an equality \(r_4^+(k)+ \operatorname {rank}R_k= s+t-1\), from which it follows \(\operatorname {rank} R_k\geq [\frac{t}{2}]\), where \(R_k\), called the Rédei matrix, denotes a \(s+t\) square matrix with coefficients in \(\mathbb{Z}/2\mathbb{Z}\) obtained by Kronecker symbols derived from \(d\) and \(p_i^*\). Based on this result, the author gives a characterization of the Rédei matrix with minimal rank, i.e., \(\operatorname {rank} R_k= [\frac{t}{2}]\). This generalizes a result of \textit{R. J. Kingan} [Can. Math. Bull. 38, 330-333 (1995; Zbl 0877.11056)].