Polynomials whose Galois groups are Frobenius groups with prime order complement
type: Item
instance of: scholarly article
title: Polynomials whose Galois groups are Frobenius groups with prime order complement
zbMATH Open document ID: 0828.12004
author: Leonardo Cangelmi
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 1996-01-07T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/93610
full work available at URL: jtnb6-2.html
review text: 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 characteriz…
reviewed by: Noriko Yui
Mathematics Subject Classification ID: 12F12
Mathematics Subject Classification ID: 12Y05
zbMATH DE Number: 754099
zbMATH Keywords: inverse Galois problem
zbMATH Keywords: resultant polynomial
zbMATH Keywords: irreducible polynomials
zbMATH Keywords: Galois groups
zbMATH Keywords: Frobenius groups
zbMATH Keywords: complement of prime order
MaRDI profile type: Publication
OpenAlex ID: W2320086395
cites work: Polynomials with Frobenius groups of prime degree as Galois groups. II
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DOI: 10.5802/JTNB.121
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description: scientific article; zbMATH DE number 754099