# Principal Ideal Domain is Unique Factorization Domain

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## Theorem

Every principal ideal domain is a unique factorization domain.

## Proof

From Element of Principal Ideal Domain is Finite Product of Irreducible Elements, each element which is neither $0$ nor a unit of a principal ideal domain has a factorization of irreducible elements.

## Sources

- 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): $\S 62$. Factorization in an integral domain