On the rank of the \(2\)-class group of \(\mathbb{Q}(\sqrt m,\sqrt d)\) where \(m=2\) or a prime \(p\equiv 1\pmod 4\) (Q2719046)

The authors determine the rank of the 2-class group of real biquadratic number fields of the form \(K= \mathbb{Q}(\sqrt{m},\sqrt{d})\) where \(m=2\) or \(m\) is a prime congruent to \(1\bmod 4\), and \(d\) is a square-free positive integer. Denoting \(k= \mathbb{Q}(\sqrt{m})\), \(t\) the number of primes in \(k\) that ramify in \(K\), and \(2^e= [E: E\cap N(K^*)],\) where \(E\) is the unit group of \(k\) and \(N(K^*)\) is the norm from \(K\) to \(k\) of the nonzero elements in \(K\), the authors use the ambiguous class number formula to establish that the rank of the 2-class group of \(K\), which we denote by rank \(H\), is equal to \(r-1-e\). The main result (stated as two theorems) can be described as follows: NEWLINENEWLINENEWLINEIf there exists an odd prime \(q\) such that \(q\equiv 3\bmod 4\) and \(q\) divides \(d\), then \(e=1\) or \(e=2\); if there does not exist an odd prime \(q\equiv 3\bmod 4\) such that \(q\) divides \(d\) then \(e=0\) or \(e=1\). NEWLINENEWLINENEWLINEIn each case the authors determine completely the value of \(e\), and thereby rank \(H\), through criteria involving congruence relations, Kronecker symbols, and biquadratic residue symbols of the primes dividing \(m\) and \(d\). The criteria is established by means of genus theory and an extensive use of the properties of the norm residue symbol. NEWLINENEWLINENEWLINEAs an application of the main result, the authors demonstrate that if \(K= \mathbb{Q}(\sqrt{p},\sqrt{d})\), \(p\equiv 1\bmod 4\), \(d\) a square-free positive integer, \(K=K^g\) where \(K^g\) is the genus field of \(K\), \(h(K)\) denotes the 2-class number of \(K\), and \(h(pd)\) denotes the 2-class number of \(Q(\sqrt{p},\sqrt{d})\), then \(h(K)= \frac 12 h(pd)\) and the 2-class group of \(K\) is trivial or cyclic. As another application of the main result, the authors examine the field \(K= \mathbb{Q}(\sqrt{p},\sqrt{p_1q_1q_2})\) such that \(p\equiv p_1\equiv q_1\equiv q_2\equiv 1\bmod 4\) and utilize the Legendre symbols of the primes \(p\), \(p_1\), \(q_1\), and \(q_2\) to determine completely when the 2-class group of \(K\) is cyclic and when the 2-class group of \(k\) is isomorphic to \(\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}\). A number of examples are given to illustrate both of these applications.