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===Unique factorization property of polynomial rings===
For proving that a polynomial ring over a unique factorization ___domain is a unique unique factorization ___domain,{{clarification needed|date=July 2016}} it suffices to consider the [[univariate]] case, as the general case may be deduced by a [[mathematical induction|recurrence]] on the number of indeterminates.
The unique factorization property is a direct consequence of [[Euclid's lemma]]: ''if an [[irreducible element]] divides a product, then it divides one of the factor. For univariate polynomials over a field, this results from [[Bézout's identity]], which results itself from [[Euclidean algorithm]].
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