Eisenstein generalized this result to his namesake integers, and Gauss generalized this theorem to the ring of complex integers.
Propositions 30-32 of Book 7, as well as Proposition 14 of Book 9, of Euclid's Elements constitute a statement of this result. Rings in which this statement holds for ideals are called Dedekind domains. Eisenstein generalized this result to his namesake integers, and Gauss generalized this theorem to the ring of complex integers. This result provides for the uniqueness of   canonical representations. Rings where this result is true include Euclidean and principal ideal domains and are generally called unique factorization domains. For ten points, identify this result from number theory which guarantees the existence of a unique prime factorization for natural numbers.
Propositions 30-32 of Book 7, as well as Proposition 14 of Book 9, of Euclid's Elements constitute a statement of this result.
fundamental theorem of arithmetic [prompt on fundamental theorem, do not accept or prompt on  fundamental theorem of [anything else] , accept unique prime factorization theorem until  unique prime factorization  is read and prompt afterwards]