On the parity of the class number of the field \(\mathbb Q(\sqrt p,\sqrt q,\sqrt r)\) (Q1385265)

The author determines the parity of the class number of the octic field \(K= \mathbb Q(\sqrt{p}, \sqrt{q}, \sqrt{r})\), where \(p,q,r\) are different primes and \(p\equiv q\equiv r\equiv 1\bmod 4\). The author establishes that if the Legendre symbols \((p/q)= (p/r)= (q/r)=1\) or \((p/q)= (q/r)= 1\), \((p/r)=-1\) then \(K\) has even 2-class number (i.e. the class number of the 2-Sylow subgroup of the ideal class group of \(K\)). If \((p/q)=1\), \((p/r)= (q/r)=-1\) then the parity of the class number of \(K\) is the same as the parity of the class number of the biquadratic field \(\mathbb Q (\sqrt{p}, \sqrt{q})\); if \((p/q)= (p/r)= (q/r)=-1\) then the parity of the class number of \(K\) is determined by a combination of specific congruence relations and related Legendre symbols. In addition, the author shows that if \((p/q)=(p/r)=(q/r)=1\), then the 2-class number of \(K\) is greater than or equal to twice the product of the 2-class numbers of the three quadratic subfields of \(K\). In order to prove these results the author utilizes the results and techniques of \textit{R. Kučera} [J. Number Theory 52, 43--52 (1995; Zbl 0852.11065)] for the biquadratic field \(\mathbb Q (\sqrt{p}, \sqrt{q})\) where \(p\) and \(q\) are different primes and \(p\equiv q\equiv 1\bmod 4\). The techniques utilized in the present paper revolve around the cyclotomic units of \(K\), the fundamental units in the quadratic subfields of \(K\), the fixed Dirichlet characters modulo \(t\) of order 4, where \(t\neq 1\) divides \(pqr\), and crossed homomorphisms from \(G= \text{Gal} (K/\mathbb Q)\) to \(K\). Specifically, the formulation for the parity of the class number depends upon units of the form \(B_{pq}= \prod_{a\in M_{pq}} ({\mathcal E}_{pq}^a- {\mathcal E}_{pq}^{-a})\), where \(M_{pq}= \{a\in \mathbb Z\mid 0<a< pq\), \((a/q)=1\}\), \(\psi_p(a)\in R_p\), where the symbols are defined as follows: Let \(E\) be the group of units in \(K= \mathbb Q (\sqrt{p}, \sqrt{q}, \sqrt{r})\), \({\mathcal E}_n= e^{\frac{2\pi i}{n} \frac{1+n}{2}}\) for any positive integer \(n\); for any prime \(t\equiv 1\bmod 4\) let \(b_t,c_t\) be integers such that \(t-1= 2^{b_t}c_t\), where \(2\nmid c_t\) and \(b_t\geq 2\), and let \(\psi_t\) denote a fixed Dirichlet character modulo \(t\) of order \(2^{b_t}\); let \(R_t= \{p_t^j\mid 0\leq j<2^{b_t-2}\}\) where \(p_t= e^{4\pi ict/t-1}\) is a primitive \(2^{b_t-1}\)th root of unity. A basic result used to establish the parity of the class number \(h\) of \(K\) states that \(h\) is even if and only if there are \(\chi_p, \chi_q, \chi_r, \chi_{pq},\chi_{qr}, \chi_{pqr}\in \{0,1\}\) such that \[ n=| {\mathcal E}_p^{\chi_p} {\mathcal E}_q^{\chi_q}{\mathcal E}_r^{\chi_r} {\mathcal B}_{pq}^{\chi_{pq}} {\mathcal B}_{pr}^{\chi_{pr}} {\mathcal B}_{qr}^{\chi_{qr}} {\mathcal B}_{pqr}^{\chi_{pqr}} |\neq 1 \] is a square in \(E\) \(({\mathcal E}_s\) denotes the fundamental unit in the quadratic field \(\mathbb Q (\sqrt{s})\) and \(\chi_s\) denotes a fixed Dirichlet character modulo \(S\) of order 4). For example, in order to establish the parity criteria when \((p/q)= (q/r)=1\), \((p/r)=-1\), the author demonstrates that a necessary condition for \(n\) being a square in \(E\) is that \(\chi_q(r)\cdot \chi_r(q)= \chi_q(p)\cdot \chi_p(q)\).