Unique Factorization
In an integral domain R, the decomposition of a nonzero noninvertible element a as a product of prime (or irreducible) factors
| a=p_1...p_n, |
(1)
|
is unique if every other decomposition of the same type has the same number of factors
| a=q_1...q_n, |
(2)
|
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)],
| 6=(1+isqrt(5))(1-isqrt(5))=2·3 |
(3)
|
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) |
(4)
|
lie both outside Z[isqrt(5)]. Furthermore,
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 DomainThis entry contributed by Margherita Barile
Explore with Wolfram|Alpha
References
Sigler, L. E. Algebra. New York: Springer-Verlag, 1976.Referenced on Wolfram|Alpha
Unique FactorizationCite this as:
Barile, Margherita. "Unique Factorization." From MathWorld--A Wolfram Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/UniqueFactorization.html