The fundamental theorem can be derived from Book VII, propositions 30, 31 and 32, and Book IX, proposition 14 of Euclid's Elements.
If two numbers by multiplying one another … 展开

Canonical representation of a positive integer
Every positive integer n > 1 can be represented in exactly one way as a product of prime powers
where p1 < p2 < ... < pk … 展开

The proof uses Euclid's lemma (Elements VII, 30): If a prime divides the product of two integers, then it must divide at least one of these integers.
Existence
It must be shown that every integer greater than 1 is either … 展开

The first generalization of the theorem is found in Gauss's second monograph (1832) on biquadratic reciprocity. This paper introduced what is now called the ring of Gaussian integers, the set of all complex numbers a + bi where a and b are integers. It is now … 展开

• Integer factorization – Decomposition of a number into a product
• Prime signature – Multiset of prime exponents in a prime factorization 展开

1. ^ Gauss (1986, Art. 16)
2. ^ Gauss (1986, Art. 131)
3. ^ Long (1972, p. 44)
4. ^ Pettofrezzo & Byrkit (1970, p. 53) 展开

• Why isn’t the fundamental theorem of arithmetic obvious?
• GCD and the Fundamental Theorem of Arithmetic at cut-the-knot.
• PlanetMath: Proof of fundamental theorem of arithmetic 展开