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. For rings more general than the complex polynomials C[x], 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.
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