The fundamental theorem of arithmetic states that every positive integer (except the number 1) can be represented in exactly
one way apart from rearrangement as a product of
one or more primes (Hardy and Wright 1979, pp. 2-3).
This theorem is also called the unique factorization theorem. The fundamental theorem of arithmetic is a corollary of the first of Euclid's
theorems (Hardy and Wright 1979).
For rings more general than the complex polynomials ,
there does not necessarily exist a unique factorization. However, a principal
ideal domain is a structure for which the proof of the unique factorization property
is sufficiently easy while being quite general and common.