Group Theory in Number Theory
- 1 School of Mathematics, Northwest University, Xi’an, Shaanxi, 710127, China
* Author to whom correspondence should be addressed.
Abstract
The theory of groups exists in many fields of mathematics and has made a great impact on many fields of mathematics. In this article, this paper first introduces the history of group theory and elementary number theory, and then lists the definitions of group, ring, field and the most basic prime and integer and divisor in number theory that need to be used in this article. Then from the definitions, step by step, Euler's theorem, Bézout's lemma, Wilson's theorem and Fermat's Little theorem in elementary number theory are proved by means of definitions of group theory, cyclic groups, and even polynomials over domains. Finally, some concluding remarks are made. Many number theory theorems can be proved directly by the method of group theory without the action of tricks in number theory. Number theory is the thinking of certain special groups (e.g., (Z,+),(Z,×)), so the methods of group theory work well inside number theory.
Keywords
Group theory, Number theory, Ring theory.
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Cite this article
Wang,M. (2023). Group Theory in Number Theory. Theoretical and Natural Science,5,9-13.
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Volume title: Proceedings of the 2nd International Conference on Computing Innovation and Applied Physics (CONF-CIAP 2023)
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