Explicit construction of certain split extensions of number fields and constructing cyclic class fields (Q1314929)

Let \(F\) be an algebraic number field. For a prime power \(n= \ell^ u\) let \(G\) be a group of the form \(G=V \cdot T\), where \(T\) is cyclic of order \(n\) with \(T\trianglelefteq G\) and \(V\) is isomorphic to a subgroup of \((\mathbb{Z}/ n\mathbb{Z})^ \times\). In the present paper the author gives an algorithm of an explicit construction of an extension \(K/F\) such that \(\text{Gal} (K/F) =G\), \(K\cap \mathbb{Q} (\zeta) =\mathbb{Q}\) in terms of the arithmetic of the subfield \(k\) of \(K\) fixed by \(T\), which relies on class field theory and the properties of Lagrange resolvents, where \(\zeta\) denotes a primitive \(n\)-th root of unity. Choose an irreducible polynomial over \(F\) with splitting field \(k'\) and its root \(\varepsilon\) with \(\text{Gal} (k'/F) \simeq H= (\mathbb{Z}/ n\mathbb{Z})^ \times\). For some square matrix \((a_{s,t})\) \((s,t\in H)\) depending only on \(n\), set \(\beta_ \nu= \prod_{t\in H} \varepsilon_ t^{a_{\nu,t}}\), where \(\varepsilon_ t\) are conjugates of \(\varepsilon\). Then \(\alpha= {1 \over n} \sum_ \nu \beta_ n^{1/n}\) generates an extension \(R/F\) of degree \(n\) and its minimal polynomial \(p(x)\) has splitting field \(K\) with \(\text{Gal} (K/F)= \text{Gal} (K/R)\cdot T\). The coefficients of \(p(x)\) are calculated from \(\beta_ \nu\) and \(\zeta\). Furthermore the ramification of primes in \(K/k\) is controlled by the ramification in \(k(\zeta, \beta_ \nu^{1/n})/ k(\zeta)\). As examples of this construction the author gives, with detailed discussions, explicit generators and their minimal polynomials for some class fields over \(k\) of that type, such as the Hilbert class field of \(\mathbb{Q} (\sqrt {-47})\), a cyclic extension over \(\mathbb{Q} (\zeta_{16}+ \zeta^ 7_{16})\) of degree 5 ramified only at 5 with \(F= \mathbb{Q}\), and a cyclic extension over \(\mathbb{Q} (\root 4\of {2})\) of degree 5 ramified only at 5 with \(F= \mathbb{Q} (\sqrt{2})\).