Hilbert class fields of real biquadratic fields
The author gives a method for constructing the Hilbert 2-class field of a real biquadratic field of the form \(K= \mathbb{Q} (\sqrt {p}, \sqrt {d})\) under certain conditions. Here \(p\equiv 1 \pmod 8\) is a prime and \(d\equiv 3\pmod 4\). Let \(k_ 0= \mathbb{Q} (\sqrt {p})\), \(k_ 1= \mathbb{Q} (\sqrt {d})\) and \(k_ 2= \mathbb{Q} (\sqrt {pd})\) denote the quadratic subfields of \(K\). If the ramified prime ideals generate the 2-class groups of \(k_ 1\) and \(k_ 2\) then it is shown that \[ E= \mathbb{Q} (\sqrt {p}, \sqrt {q_ 1}, \dots, \sqrt {q_ n}, \sqrt {\alpha_ 1}, \dots, \sqrt {\alpha_ m}) \] is a subfield of the Hilbert class field of \(K\) with \([E: K]= 2^{m+n -1}\). Here \(d= q_ 1 q_ 2\dots q_ n\) is the product of distinct primes and \(({p \over {q_ i}})=+1\) if and only if \(i\leq m\). The class number \(h_ 0\) of \(k_ 0\) is odd and \(4q_ i^{h_ 0}= x_ i^ 2- py^ 2_ i\) for some integers \(x_ i\) and \(y_ i\) when \(i\leq m\). Here \(\alpha_ i= (x_ i+ y_ i \sqrt {p})/2\). It is proved that if the ramified prime ideals of \(k_ 1\) lying over \(q_ i\) for \(1\leq i\leq m\) together with the ideal over 2 generate the 2-class group of \(k_ 1\), then \(E\) is the Hilbert 2-class field of \(K\). Finally, an example is given showing how to obtain the Hilbert 2-class field of a real biquadratic field \(K\) with class group \(\mathbb{Z}_ 4\times \mathbb{Z}_ 4\).
scientific article; zbMATH DE number 714116