Field extensions defined by power compositional polynomials
type: Item
instance of: scholarly article
title: Field extensions defined by power compositional polynomials
zbMATH Open document ID: 1486.12004
DOI: 10.35834/2021/3302163
author: Chad Awtrey
author: James R. Beuerle
author: Hanna Noelle Griesbach
published in: Missouri Journal of Mathematical Sciences
publication date: 2022-03-18T00:00:00Z
review text: Let \(k>1\) and \(m>0\) be integers, \(F\) a field of characteristic \(0\), and \(f(x)\in F[x]\) an irreducible polynomial of degree \(km\). Let \(L = F(\alpha)\), where \(\alpha\) is a root of \(f(x)\). The aim of this paper (under review) is to study the following problem formulated in the light of the results of \textit{C. Awtrey} and \textit{P. Jakes} [Can. Math. Bull. 63, No. 3, 670--676 (2020; Zbl 1458.11173)], \textit{J. W. Jones} and \textit{D. P. Roberts} [J. Number Theory 128, No. 6, 1410--1429 (2008; Zbl 1140.11056)], \textit{M.-c. Kang} [Am. Math. Mon. 107, No. 3, 254--256 (2000; Zbl 0990.12001)]: when does there exist an irreducible polynomial \(g(x^k)\in F[x]\) such that \(L\) is isomorphic to \(F[x]/(g(x^k))\), and how do we construct such a polynomial \(g(x)\), if it exists? If \(F\) contains the \(k\)-th roots of unity, then Kummer theory provides an answer. More precisely, the author has shown using the classical result from Kummer theory that \(L/F\) can be defined by \(g(x^k)\) if and only if \(L/F\) has an automorphism \(\sigma\) of order \(k\) such that the fixed field of the cyclic group generated by \(\sigma\) contains the \(k\)-th roots of unity. Furthermore, he gave an algorithm, based on \(\sigma\), which allows the construction of \(g(x^k)\). This approach is not new; for example, the special case \(k=2\) was used in [\textit{J. W. Jones} and \textit{D. P. Roberts}, J. Number Theory 128, No. 6, 1410--1429 (2008; Zbl 1140.11056), Section 2] to construct irreducible even polynomials of degree 8 defining octic 2-adic fields with a quartic subfield. In particular, \(L/F\) is defined by an irreducible even polynomial in \(F[x]\) of degree \(2m\) if and only if \(L/F\) has an even number of automorphisms. However, the analogous statement is not necessarily true for \(k>2\). If \(L\) does not contain the \(k\)-th roots of unity, the main result is to explicitly describe the extensions \(L/F\) which are defined by irreducible polynomials of the f…
reviewed by: El Hassane Fliouet
Mathematics Subject Classification ID: 12F05
Mathematics Subject Classification ID: 12-08
Mathematics Subject Classification ID: 20B35
Mathematics Subject Classification ID: 11R32
zbMATH DE Number: 7492924
zbMATH Keywords: Galois groups
zbMATH Keywords: sextic extensions
zbMATH Keywords: power compositional
zbMATH Keywords: automorphisms
zbMATH Keywords: roots of unity
zbMATH Keywords: radical extensions
zbMATH Keywords: Kummer theory
describes a project that uses: GAP
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.35834/2021/3302163
OpenAlex ID: W3215744144
cites work: Galois Groups of Even Sextic Polynomials
cites work: Centralizers of transitive permutation groups and applications to Galois theory
cites work: Q5094817
cites work: The transitive groups of degree up to eleven+
cites work: Quartic fields and radical extensions.
cites work: Q3139838
cites work: Q3348991
cites work: Q4882944
cites work: Q4423053
cites work: On Invariant Polynomials and Their Application in Field Theory
cites work: The irreducibility of power compositional sextic polynomials and their Galois groups
cites work: Octic 2-adic fields
cites work: Cubic Fields and Radical Extensions
cites work: An Elementary Test for the Galois Group of a Quartic Polynomial
cites work: Generating subfields
description: scientific article; zbMATH DE number 7492924