Analysis of the Related Corollaries, Concepts, and Impacts of the Fundamental Theorem of Arithmetic in Number Theory and Abstract Algebra

Wenbo Tang

doi:10.54254/2753-8818/2026.CH30671

Wenbo Tang^*

Soochow Foreign Language School

^* Corresponding author: wenbotang@appmail.cn

Published: 18 December 2025

DOI: https://doi.org/10.54254/2753-8818/2026.CH30671

Abstract

The Fundamental Theorem of Arithmetic (FTA), first formalized in Euclid’s Elements around 300 BCE, established the foundation for classical number theory, composed around 300 BCE. The principle of unique factorization later became central to the rise of modern mathematics. In the mid-19th century, mathematicians such as J.W.R. Dedekind and D. Hilbert extended number-theoretic questions into quadratic fields and rings of algebraic integers, creating the foundations of algebraic number theory. Steinitz’s work in the early twentieth century of 1910 further generalized algebraic structures, marking the beginning of abstract algebra as an independent field. The purpose of this essay is to examine the Fundamental Theorem of Algebra's proof. and applies it to several representative problems in elementary number theory. It then extends to related corollaries and conceptual developments of unique factorization, including notable cases in non-unique factorization rings where the property does not hold. Finally, it introduces approaches grounded in properties of the FTA that have been applied to the ongoing study of major open problems, including the Goldbach Conjecture. Overall, through both review and mathematical analysis, the paper shows that the structural foundation provided by the FTA underlies the verification and proof of many of the most difficult results and open conjectures in mathematics, including Fermat’s Last Theorem and the Goldbach Conjecture.

Keywords: FTA , Algebraic Number Theory , Abstract Algebra , Dedekind's Unique Factorization Theorem , Goldbach Conjecture

TNS Vol.151

ISSN (Print): 2753-8818

ISSN (Online): 2753-8826

ISBN (Print): 978-1-80590-559-2

ISBN (Online): 978-1-80590-560-8

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Received: 7 November 2025

Published: 18 December 2025

DOI: https://doi.org/10.54254/2753-8818/2026.CH30671

Article Type: Research Article

Cite this article

Tang,W. (2025). Analysis of the Related Corollaries, Concepts, and Impacts of the Fundamental Theorem of Arithmetic in Number Theory and Abstract Algebra. Theoretical and Natural Science,151,107-115.

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Volume Title: TNS Vol.151

Part of Series: Theoretical and Natural Science

ISSN: 2753-8818 (Print) / 2753-8826 (Online)

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