Theoretical and Natural Science
- The Open Access Proceedings Series for Conferences
Proceedings of the 2nd International Conference on Computing Innovation and Applied Physics (CONF-CIAP 2023)
Series Vol. 5 , 25 May 2023
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Reducibility of quartic polynomials and its relation to Galois groups
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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
- © 2023 The Author(s)
- 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