A nonzero polynomial over a field is square-free if no square of a nonconstant irreducible polynomial divides it. Equivalently, in its factorization of polynomials, every irreducible polynomial factor occurs with exponent one. For example, is square-free over the real numbers, whereas is not. Square-freeness over a given field concerns factors in that field; over the real numbers or complex numbers, it is equivalent to having no repeated polynomial roots after splitting over the complex numbers.
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A square-free polynomial is a polynomial that does not have any repeated roots in its factorization over a given field or ring. In other words, if a polynomial is expressed in its factored form, none of the factors appear more than once. For example, consider the polynomial \( P(x) = x^2 - 2x \).