On the density of the splitting of 2 in cubic fields (Q1327140)

Let \(\Lambda\) denote the set of irreducible monic cubic polynomials \(f\in \mathbb{Z}[x]\), and let \(\Lambda(N)\) be the subset consisting of those \(f\) for which the absolute values of the coefficients do not exceed \(N\). The density of a set \(X\) contained in \(\Lambda\) is defined as \(d(X)= \lim_{N\to\infty} | X\cap \Lambda(N)|/ |\Lambda (N)|\). If \(\alpha\) is a zero of an \(f\in\Lambda\), there are five possible splitting types of the prime 2 in \(\mathbb{Q}(\alpha)\). For each splitting type, the authors compute the explicit value of the corresponding density.