The Fundamental Theorem of Arithmetic (FTA), first formalized in Euclids 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. Steinitzs 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 Fermats Last Theorem and the Goldbach Conjecture.
Wai Shing Tang (Thu,) studied this question.