In mathematics, an element "r" of a unique factorization domain "R" is called square-free if it is not divisible by a non-trivial square. That is, every "s" such that s^2mid r is a unit of "R".

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:r=p_1p_2cdots p_nThen "r" is square-free if and only if the primes "pi" 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).

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

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