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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Vydáno v:	Indagationes mathematicae Ročník 33; číslo 2; s. 482 - 493
Hlavní autoři:	Solanilla, Leonardo, Barreto, Erick S., Morales, Viviana
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	Amsterdam Elsevier B.V 01.03.2022 Elsevier Science Ltd
Témata:	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
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Shrnutí:	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.
Bibliografie:	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