In an integral domain , the decomposition of a nonzero noninvertible element as a product of prime (or irreducible) factors

is unique if every other decomposition of the same type has the same number of factors

and its factors can be rearranged in such a way that for all indices , and differ by an invertible factor.

The prime factorization of an element, if it exists, is always unique, but this does not apply, in general, to irreducible factorizations: in the ring ,

are two different irreducible factorizations, none of which is prime. 2 is not a prime element in , since it does not divide either of the factors of the middle expression. In fact

lie both outside . Furthermore,

which shows that is not prime either.

An integral domain where every nonzero noninvertible element admits a unique irreducible factorization is called a unique factorization domain.

See also

Fundamental Theorem of Arithmetic, Unique Factorization Domain

This entry contributed by Margherita Barile

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References

Sigler, L. E. Algebra. New York: Springer-Verlag, 1976.

Referenced on Wolfram|Alpha

Unique Factorization

Cite this as:

Weisstein, Eric W., with contributions by Margherita Barile. "Unique Factorization." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/UniqueFactorization.html

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