{{no footnotes, date=December 2015 In

mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, a square-free element is an element ''r'' of a

unique factorization domain In mathematics, a unique factorization domain (UFD) (also sometimes called a factorial ring following the terminology of Bourbaki) is a ring in which a statement analogous to the fundamental theorem of arithmetic holds. Specifically, a UFD is ...

''R'' that is not

divisible In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a '' multiple'' of m. An integer n is divisible or evenly divisibl ...

by a non-trivial square. This means that every ''s'' such that

s^2\mid r

is a unit of ''R''.

Alternate characterizations

Square-free elements may be also characterized using their prime decomposition. The unique factorization property means that a non-zero non-unit ''r'' can be represented as a product of

prime element In mathematics, specifically in abstract algebra, a prime element of a commutative ring is an object satisfying certain properties similar to the prime numbers in the integers and to irreducible polynomials. Care should be taken to distinguish ...

s :

r=p_1p_2\cdots p_n

Then ''r'' is square-free if and only if the primes ''p_i'' are pairwise non-associated (i.e. that it doesn't have two of the same prime as factors, which would make it divisible by a square number).

Examples

Common examples of square-free elements include square-free integers and square-free polynomials.

References

*David Darling (2004) ''The Universal Book of Mathematics: From Abracadabra to Zeno's Paradoxes'' John Wiley & Sons *Baker, R. C. "The square-free divisor problem." The Quarterly Journal of Mathematics 45.3 (1994): 269-277. Ring theory

Alternate characterizations

Examples

See also

References