Polynomials whose Galois groups are Frobenius groups with prime order complement (Q1805364)

An effective characterization theorem is proved for integral monic irreducible polynomials \(f\) of arbitrary degree \(n\) whose Galois groups over \(\mathbb{Q}\) are Frobenius groups with kernel of order \(n\) and complement of prime order \(p\) with \(p \mid n - 1\). This generalizes the results of \textit{A. Bruen}, \textit{C. Jensen} and \textit{N. Yui} [J. Number Theory 24, 305-359 (1986; Zbl 0598.12009)]. Let \(L/ \mathbb{Q}\) be a normal extension and let \(K\) be a subfield of \(L\) such that \([L : K] = n\), \(K/ \mathbb{Q}\) is a cyclic extension of prime degree \(p\) and \(p \nmid n\). Put \(G = \text{Gal} (L/ \mathbb{Q})\) and \(N = \text{Gal} (L/K)\). Then \(G = N \rtimes_\theta Z_p\) is the semidirect product of the cyclic group \(Z_p = \langle \tau \rangle\) of order \(p\) and \(N\), where \(\theta\) is an automorphism of \(N\) with the action \(\tau x = \theta (x) \tau\) for \(x \in N\). It is easy to see that \(G\) is the Frobenius group \(\text{Fr} (N,p)\) (with kernel \(N\) and complement \(Z_p)\) if and only if \(\text{Fix} (\theta) = 1\). Let \(f \in \mathbb{Z} [x]\) be a monic irreducible polynomial of degree \(n\) and let \(G = \text{Gal} (f/ \mathbb{Q})\) and \(L = \text{spl} (f)\) be the splitting field of \(f\) over \(\mathbb{Q}\). The characterization theorem is given as follows: Theorem. Let \(p\) be a rational prime dividing \(n - 1\). Then \(G = \text{Fr} (N,p)\) if and only if the following conditions hold: (1) \(p \mid [L : \mathbb{Q}]\). (2) there is an extension \(k/ \mathbb{Q}\), cyclic of degree \(p\), such that \(f\) is irreducible and normal over \(k\). (3) there exists a rational prime \(q\), unramified in \(L\), such that \(f \text{mod} q\) is of (cycle) type \(1.p^{(n - 1)/p}\). An effective version of the characterization theorem is formulated assuming the Generalized Riemann Hypothesis (GRH) and using generalized resultant polynomials. The effective version provides a way of verifying the conditions given in the characterization theorem. The characterization theorem is then tested on examples of polynomials with Frobenius Galois groups \(\text{Fr} (\mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_7, 3)\) and \(\text{Fr} (\mathbb{Z}_7 \times \mathbb{Z}_7,3)\).