{{no footnotes, date=December 2015 In mathematics, 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 divisible by ...

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

s^2\mid r

is a

unit Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in a theatrical presentation Music * ''Unit'' (alb ...

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 pri ...

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 integer In mathematics, a square-free integer (or squarefree integer) is an integer which is divisible by no square number other than 1. That is, its prime factorization has exactly one factor for each prime that appears in it. For example, is square-f ...

s and

square-free polynomial In mathematics, a square-free polynomial is a polynomial defined over a field (or more generally, an integral domain) that does not have as a divisor any square of a non-constant polynomial. A univariate polynomial is square free if and only if i ...

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