A brief account of Klein’s icosahedral extensions

We present an alternative relatively easy way to understand and determine the zeros of a quintic polynomial whose Galois group is isomorphic to the group of rotational symmetries of a regular icosahedron. The extensive algebraic procedures of Klein in his famous Vorlesungen über das Ikosaeder und di...

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Veröffentlicht in:Indagationes mathematicae Jg. 33; H. 2; S. 482 - 493
Hauptverfasser: Solanilla, Leonardo, Barreto, Erick S., Morales, Viviana
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Amsterdam Elsevier B.V 01.03.2022
Elsevier Science Ltd
Schlagworte:
Algebra
Group theory
Hypergeometric functions
Icosahedrons
Mathematical analysis
Polynomials
Polynomials in real and complex fields, Location of zeros (algebraic theorems)
Real polynomials, Location of zeros
Zeros of polynomials, rational functions, and other analytic functions of one complex variable
Real polynomials, Location of zeros
Zeros of polynomials, rational functions, and other analytic functions of one complex variable
Polynomials in real and complex fields, Location of zeros (algebraic theorems)
ISSN:0019-3577, 1872-6100
Online-Zugang:Volltext
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Zusammenfassung:We present an alternative relatively easy way to understand and determine the zeros of a quintic polynomial whose Galois group is isomorphic to the group of rotational symmetries of a regular icosahedron. The extensive algebraic procedures of Klein in his famous Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom fünften Grade are here shortened via Heymann’s theory of resolvents. Also, we give a complete explanation of the so-called icosahedral equation and its solution in terms of Gaussian hypergeometric functions. As an innovative element, we construct this solution by using algebraic transformations of hypergeometric series.
Bibliographie:ObjectType-Article-1
SourceType-Scholarly Journals-1
ObjectType-Feature-2
content type line 14
ISSN:0019-3577
1872-6100
DOI:10.1016/j.indag.2021.10.002