Square-free element

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

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 Then 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.