The generalized Rédei-matrix (Q957914)

Let \(F\) be a number field with odd class number and \(E = F(\sqrt{\delta}\,)\) a quadratic extension. The main result proved in this article is the formula \[ \text{rk}_4\;\text{Cl}(E) = m - 1 - \text{rk}\;R_{E/F} \] for the \(4\)-rank of the class group Cl\((E)\), where \(m\) denotes the number of primes ramified in \(E/F\), and \(R_{E/F}\) is the generalized Rédei matrix formed from local Hilbert symbols. This generalizes the corresponding result for quadratic extensions due to \textit{L. Rédei} [J. Reine Angew. Math. 171, 55--60 (1934; Zbl 0009.05101)], (see [\textit{P. Stevenhagen}, ``Rédei-matrices and applications'', Lond. Math. Soc. Lect. Note Ser. 215, 245--259 (1995; Zbl 0830.11039)] for a modern presentation). For defining the generalized Rédei matrix, let \({\mathfrak p}_1, \dots, {\mathfrak p}_m\) denote the (finite and infinite) primes ramified in \(E/F\), and write \({\mathfrak p} O_E = {\mathfrak P}^2\) for the finite primes \({\mathfrak p}_1, \dots, {\mathfrak p}_n\). Let \(\upsilon_1, \dots, \upsilon_r\) denote a basis of the unit group of the ring of integers in \(F\). This basis can be chosen in such a way that the first \(\ell\) units are norms of elements from \(E\), and that \((O_F^\times : O_F^\times \cap N E) = 2^{r - \ell}\). The \(2\)-class group of \(E\) is generated by strongly ambiguous ideal classes \([{\mathfrak P}_1], \dots, [{\mathfrak P}_n]\) and by weakly ambiguous ideal classes \([{\mathfrak B}_1], \dots, [{\mathfrak B}_\ell]\), with \(\ell\) as above. Define elements \(a_i, b_j \in F\) by setting \((a_i) = ({\mathfrak P}_i^{-h} \sigma({\mathfrak P}_1)^{-h})\) for \(1 \leq i \leq n\) and \((b_j) = ({\mathfrak B}_i^{-h} \sigma({\mathfrak B}_1)^{-h})\) for \(1 \leq j \leq \ell\), where \(\sigma\) is the nontrivial automorphism of \(E/F\). The generalized Rédei matrix is an \(m \times (n+r)\)-matrix whose entries are the Hilbert symbols \((a_i,\delta)_{{\mathfrak p}_j}\), \((b_i,\delta)_{{\mathfrak p}_j}\), and \((\upsilon_k,\delta)_{{\mathfrak p}_j}\) for \(\ell+1 \leq k \leq r\). The last section contains applications to quadratic and biquadratic number fields.