Theoretical and Natural Science

- The Open Access Proceedings Series for Conferences


Theoretical and Natural Science

Vol. 5, 25 May 2023


Open Access | Article

Reducibility of quartic polynomials and its relation to Galois groups

Jinhe Huang * 1 , Yicheng Ai 2 , Yuhao Wen 3
1 The Experimental High School Attached to Beijing Normal University
2 Tsinglan School
3 University of London

* Author to whom correspondence should be addressed.

Theoretical and Natural Science, Vol. 5, 439-447
Published 25 May 2023. © 2023 The Author(s). Published by EWA Publishing
This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Citation Jinhe Huang, Yicheng Ai, Yuhao Wen. Reducibility of quartic polynomials and its relation to Galois groups. TNS (2023) Vol. 5: 439-447. DOI: 10.54254/2753-8818/5/20230277.

Abstract

This paper deals with special classes of quartic polynomials and properties pertaining to their Galois groups and reducibility over certain fields. The existence of quartic polynomials irreducible over Q but reducible over every prime field is first proven, after which criteria are established for the Galois group of polynomials with this property. By constructing classes of V4-generic polynomials and comparing them with criteria put forth in previous studies for determining polynomials with this property, it can be shown that a polynomial of the biquadratic form x4 + ax2 + b has this property if and only if it can be written as x^4 - 2(u + v)x^2 + 〖(u -v)〗^2 with u, v ∈Q such that none of u, v, or uv can be expressed as ratio of two squares, and 2(u+v),(u−v)2 ∈ Z . The general form for biquadratic polynomials irreducible over Q and reducible modulo every integer n is found to have a general form similar to this one.

Keywords

quartic polynomials, Galois groups, math

References

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Data Availability

The datasets used and/or analyzed during the current study will be available from the authors upon reasonable request.

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Volume Title
Proceedings of the 2nd International Conference on Computing Innovation and Applied Physics (CONF-CIAP 2023)
ISBN (Print)
978-1-915371-53-9
ISBN (Online)
978-1-915371-54-6
Published Date
25 May 2023
Series
Theoretical and Natural Science
ISSN (Print)
2753-8818
ISSN (Online)
2753-8826
DOI
10.54254/2753-8818/5/20230277
Copyright
25 May 2023
Open Access
This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited

Copyright © 2023 EWA Publishing. Unless Otherwise Stated