Classifying Galois groups of an orthogonal family of quartic polynomials

Pradipto Banerjee and Ranjan Bera
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 27, 2021, Number 2, Pages 172—190
DOI: 10.7546/nntdm.2021.27.2.172-190
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Authors and affiliations

Pradipto Banerjee
Department of Mathematics, Indian Institute of Technology, Hyderabad
Kandi, Telangana 502285, India

Ranjan Bera
Department of Mathematics, Indian Institute of Technology, Hyderabad
Kandi, Telangana 502285, India

Abstract

We consider the quartic generalized Laguerre polynomials L_{4}^{(\alpha)}(x) for \alpha \in \mathbb Q. It is shown that except \mathbb Z/4\mathbb Z, every transitive subgroup of S_{4} appears as the Galois group of L_{4}^{(\alpha)}(x) for infinitely many \alpha \in \mathbb Q. A precise characterization of \alpha\in \mathbb Q is obtained for each of these occurrences. Our methods involve the standard use of resolvent cubics and the theory of p-adic Newton polygons. Using these, the Galois group computations are reduced to Diophantine problem of finding integer and rational points on certain curves.

Keywords

  • Galois groups
  • Quartic polynomials
  • Generalized Laguerre polynomials
  • Newton polygons
  • Diophantine equations

2020 Mathematics Subject Classification

  • 11R32
  • 11C08
  • 33C45

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Cite this paper

Banerjee, P., & Bera, R. (2021). Classifying Galois groups of an orthogonal family of quartic polynomials. Notes on Number Theory and Discrete Mathematics, 27(2), 172-190, doi: 10.7546/nntdm.2021.27.2.172-190.

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