On the lower bound of the rank of the 2-class group of certain multiquadratic fields (Q2882480)

Let \(p_1\), \(p_2\), \(p_3\) and \(q\) be prime numbers such that \(p_1 \equiv p_2 \equiv p_3 \equiv - q \equiv 1\pmod 4\), \(k = \mathbb Q(\sqrt{p_1}, \sqrt{p_2}, \sqrt{p_3}, \sqrt{q})\) and \(Cl_{2}(k)\) be the 2-class group of \(k\). Fröhlich has shown that \(Cl_{2}(k)\) can never be trivial. In this paper, the author gives an extension of this result by proving that the rank of \(Cl_{2}(k)\) is greater than \(2\). In particular, the author proves that there exist infinitely many fields \(k\) in which the rank of \(Cl_{2}(k)\) is equal to \(2\).