{{no footnotes|date=December 2015}} In mathematics, a '''square-free element''' is an element ''r'' of a unique factorization domain ''R'' that is not divisible by a non-trivial square. This means that every ''s'' such that <math>s^2\mid r</math> 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 elements :<math>r=p_1p_2\cdots p_n</math> Then ''r'' is square-free if and only if the primes ''p<sub>i</sub>'' 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.
==See also== *Prime number ==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. Category:Ring theory