This paper proved the Fundamental Theorem of Arithmetic, which asserts the existence and uniqueness of prime factorization for every integer greater than 1, and extends it to all integers including the negatives. The proof is solely based on the ring axioms, order axioms and the well-ordering principle. After establishing the basic arithmetic operations from these axioms, the main proof is completed by defining canonical factorization, proving a key result known as the fundamental lemma (if p prime divides ab, then p divides a or b), and then demonstrating both the existence and uniqueness of prime factorizations. This proof broke free from the use of the Euclidean algorithm, Bézout’s theorem, and mathematical induction—methods commonly employed in previous proofs. By doing so, it provides a new insight into the structure of the integer ring and how “fundamental” is the Fundamental Theorem of Arithmetic.

Ke Liang. Establishing the Fundamental Theorem of Arithmetic. Academic Journal of Mathematical Sciences (2025) Vol. 6, Issue 1: 15-26. https://doi.org/10.25236/AJMS.2025.060103.

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