Unique Factorization

In an integral domain R, the decomposition of a nonzero noninvertible element a 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 i, p_i and q_i 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 Z[isqrt(5)],


are two different irreducible factorizations, none of which is prime. 2 is not a prime element in Z[isqrt(5)], since it does not divide either of the factors of the middle expression. In fact

 1/2(1+isqrt(5))=1/2+1/2isqrt(5) and 1/2(1-isqrt(5))=1/2-1/2isqrt(5)

lie both outside Z[isqrt(5)]. Furthermore,

 2/(1+isqrt(5))=1/3-i1/3sqrt(5), and 3/(1+isqrt(5))=1/2-1/2isqrt(5),

which shows that 1+isqrt(5) 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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Sigler, L. E. Algebra. New York: Springer-Verlag, 1976.

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Unique Factorization

Cite this as:

Barile, Margherita. "Unique Factorization." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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