Relations between the narrow 4-class ranks of quadratic number fields (Q2729684)

Given the quadratic number field \(k= \mathbb{Q}(\sqrt{m})\), \(m\) a square-free rational integer, \(l\) a prime number not dividing \(m\), \(C^+(k)\) the narrow ideal class group of \(k\), and \(r_4^+(k)= r_4^+(m)\) its 4-rank, the author establishes some relations between \(r_4^+(m)\) and \(r_4^+ (\pm ml)\). The basic technique utilized revolves around matrix computations to compute the rank of the Rédei matrix \(A_k\), which is obtained from the Kronecker symbols of the prime discriminants dividing the discriminant \(d\) of \(k\). If \(d= p_1^*\dots p_t^*\) is the unique decomposition of \(d\) into a product of prime discriminants, then \(r_4^+(k)= t-1- \text{rank } A_k\). By comparing the Rédei matrices \(A_m\) and \(A_{\pm ml}\) for a number of different congruent conditions on \(m\) and Kronecker symbols of type \((\frac lp)\), \((\frac{l^*}{p})\), and \((\frac{-l}{p})\) where \(p\) is a prime dividing \(m\), the author obtains inequalities of the following three forms: NEWLINE\[NEWLINE\begin{aligned} r_4^+(m) &\leq r_4^+(\pm ml)\leq r_4^+(m)+1;\\ r_4^+(m) &\leq r_4^+(-ml)\leq r_4^+(m)+1;\\ r_4^+(m)-1 &\leq r_4^+(-ml)\leq r_4^+(m)+1. \end{aligned}NEWLINE\]NEWLINE{}.