Sextic reciprocal monogenic dihedral polynomials
type: Item
instance of: scholarly article
title: Sextic reciprocal monogenic dihedral polynomials
zbMATH Open document ID: 1487.11095
published in: The Ramanujan Journal
publication date: 2021-12-03T00:00:00Z
review text: A polynomial \(f(x)\in\mathbb{Z}[x]\) with integer coefficients is said to be \textit{reciprocal} (elsewhere \textit{self-reciprocal}, or \textit{palindromic} or \textit{symmetrical})), if \(f(1/x)=f(x)/x^{deg(f)}\). A polynomial \(f\) is called \textit{monogenic}, if \(f\) is irreducible over the rationals and \(\mathbb{Z}_K=\mathbb{Z}[\theta]\), where \(\theta\) is a root of \(f\) and \(K=\mathbb{Q}(\theta)\); equivalently, the respective discriminants coincide: \(\Delta(f)=\Delta(K)\). The central result is as follows: Let Gal(\(f\)) denote the Galois group over the rationals \(\mathbb{Q}\) of the polynomial \(f(x)\in\mathbb{Z}[x]\). Denote the dihedral group of order \(2n\) by \(D_n\). Then the following hold: \begin{itemize} \item[1.] Let \(q\neq 7\) be a prime such that \(q\not\equiv\pm 1\) (mod 7). Then there exist infinitely many primes \(p\) such that the polynomial \(f(x)=x^6+x^5+(pq+1)x^4+(2pq+1)x^3+(pq+1)x^2+x+1 \) is monogenic. Furthermore, if \(q\equiv 3\) (mod 7) or \(q\equiv 5\) (mod 7), then \(\mathrm{Gal}(f)\cong D_6\). \item[2.] There are infinitely many primes \(p\) such that \(f(x)=x^6+px^3+1\) is monogenic and \(\mathrm{Gal}(f)\cong D_6\). \item[3.] There exist infinitely many primes \(p\) such that \(f(x)=x^6+3x^5+(p+6)x^4+(2p+7)x^3+(p+6)x^2+3x+1\) is monogenic and \(\mathrm{Gal}(f)\cong D_3\). \end{itemize} The emphasis in all three parts of the result is on the existence of infinitely many primes. The appropriately long proof is aided by computations in computer packages MAGMA, Maple, and Sage.
Mathematics Subject Classification ID: 11R04
Mathematics Subject Classification ID: 11R09
Mathematics Subject Classification ID: 12F05
Mathematics Subject Classification ID: 05E05
Mathematics Subject Classification ID: 11R21
zbMATH DE Number: 7438418
zbMATH Keywords: reciprocal polynomial
zbMATH Keywords: self-reciprocal polynomial
zbMATH Keywords: palindromic polynomial
zbMATH Keywords: symmetric polynomial
zbMATH Keywords: monogenic polynomial
zbMATH Keywords: discriminant of a polynomial
zbMATH Keywords: irreducible polynomial
zbMATH Keywords: Galois group
zbMATH Keywords: dihedral group
zbMATH Keywords: Chebyshev polynomial of the first kind
author: Lenny Jones
reviewed by: Radoslav M. Dimitrić
describes a project that uses: Magma
describes a project that uses: Maple
describes a project that uses: SageMath
MaRDI profile type: Publication
full work available at URL: s11139-020-00310-w
OpenAlex ID: W3046095858
cites work: Symmetric polynomials in the symplectic alphabet and the change of variables \(z_j = x_j + x_j^{-1}\)
cites work: The transitive groups of degree up to eleven+
cites work: Advanced Topics in Computional Number Theory
cites work: Galois theory
cites work: On a relationship between Chebyshev polynomials and Toeplitz determinants
cites work: \(A_4\)-sextic fields with a power basis
cites work: The Factorization of the Cyclotomic Polynomials mod p
cites work: The irreducibility of power compositional sextic polynomials and their Galois groups
cites work: Square-free values of \(f(p)\), \(f\) cubic
cites work: Q4092051
cites work: Lifting monogenic cubic fields to monogenic sextic fields
cites work: Q4254917
cites work: The ABC conjecture, arithmetic progressions of primes and squarefree values of polynomials at prime arguments
cites work: \(\text{PSL}(2,5)\) sextic fields with a power basis
cites work: Q5690489
DOI: 10.1007/S11139-020-00310-W
description: scientific article; zbMATH DE number 7438418