The 4-class ranks of quadratic fields (Q789436)

Let \(K\) be a quadratic extension field of the rational numbers \(\mathbb Q\). It is well known that the rank of the 2-class group of \(K\) (in the narrow sense) is \(t-1\), where \(t\) is the number of primes that ramify in \(K/\mathbb Q\). To obtain a deeper understanding of the structure of the 2- class group of \(K\), one may next consider the 4-class rank \(R_ K\) of \(K\), which satisfies \(0\leq R_ K\leq t-1\). An interesting question is the following: how likely is \(R_ K=0\), or \(R_ K=1\), or \(R_ K=2\), etc.? This paper answers that question by developing algorithms for computing densities of the 4-class ranks of imaginary quadratic fields and of real quadratic fields.