The Unique Factorization Theorem states that every element in an integral domain can be factored uniquely into irreducible elements, up to order and units. This property is crucial because it allows for a systematic way to understand the structure of numbers or ideals within a number system, connecting seamlessly with the concepts of ideals in rings and specific number systems like Gaussian and Eisenstein integers, where the uniqueness of factorization plays a significant role in their arithmetic properties.
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