Theoretical and Natural Science
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Theoretical and Natural Science
Vol. 13, 30 November 2023
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The interconnection between local and global m-th powers and Grunwald-Wang theorem
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Abstract
The relation between local and global solution of an equation can be discussed with the method of class field theory and algebraic number theory. In this piece of writing, the author will introduce the behavior of both local and global m-th power in some specific number field. Of course, the result in this paper can be extended into the function field, but it will not be involved in this paper. This paper will prove that if k(ω_(2^t ))/k is cyclic then the Local m-th powers everywhere is equivalent to the global m-th power. In the Non-cyclic case this decomposition becomes P(m,S)=k^m∪δk^m. This paper will also prove some useful propositions in topological group theory, which will be used in the proof of Grunwald-Wang theorem. Grunwald-Wang theorem states that we can find a cyclic extension with given local behaviors. To describe the extension, this paper combines character theory with a topological group, one can depict the cyclic extension. This theorem can be used in the further exploration of central simple algebra.
Keywords
Local And Global M-Th Powers, Cyclotomic Field, Algebraic Number Theory, Class Field Theory, Grunwald-Wang Theorem
References
1. ARTIN, E. AND TATE, J. 1961. Class field theory. Harvard, Dept. of Mathematics. Notes from the Artin-Tate seminar on class field theory given at Princeton University 1951–52. Reprinted 1968, 1990; second edition AMS Chelsea Publishing, 2009.
2. Milne, J. S. Class Field Theory. www.jmilne.org/math/. 2020:31-32,84
3. Neukirch, J. Class Field Theory, volume 280 of Grundlehren der Mathematishen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin. 1986
4. Lang, S. Algebraic Number Theory. Addison-Weysley Publishing Co.,Inc., Reading, Mass.-London-Don Mills, Ont. 1970
5. Milne, J, S. v2.00 (March 17, 2008). Corrected, revised, and expanded; https://www.jmilne.org/math/CourseNotes/CFT.pdf
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 3rd International Conference on Computing Innovation and Applied Physics
- ISBN (Print)
- 978-1-83558-189-6
- ISBN (Online)
- 978-1-83558-190-2
- Published Date
- 30 November 2023
- Series
- Theoretical and Natural Science
- ISSN (Print)
- 2753-8818
- ISSN (Online)
- 2753-8826
- DOI
- 10.54254/2753-8818/13/20240800
- Copyright
- 30 November 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