Simplest cubic fields (Q1911198)

Let \(a\) be a natural number, \(f_a = x^3 - ax^2 - (a + 3)x - 1\), and let \(K\) be the cyclic cubic field generated by a zero \(\alpha\) of \(f_a\). Suppose that \(a^2 + 3a + 9\) is squarefree. The main result is: For all \(\gamma \in \mathbb{Z} [\alpha]\), either \(|N_\gamma |\geq 2a + 3\), or \(\gamma\) is associated to a rational integer. If \(|N \gamma |= 2a + 3\), then \(\gamma\) is associated to a conjugate of \(\alpha - 1\). Using this result, the authors construct unramified extensions \(L/K\) with \(\text{Gal} (L/K) \simeq C_2 \times C_2\) in certain cases.