  It is conjectured that all separable polynomials with integers coefficients,
satisfying some local conditions, take infinitely many square free values on
integer arguments. But not a single polynomial of degree greater than $3$ is
proven to exhibit this property. In this article, we propose a method to show
that ``cyclotomic polynomials $\Phi_{\ell}(X)$ take square free values with
positive proportion". Following this method, conditionally, we do prove the
Square-free conjecture for all cyclotomic polynomials. The method is applicable
to some other class of polynomials as well. Moreover we show that the method
readily succeeds for quadratic polynomials to give unconditional result. In the
appendix, we give an elementary proof of the square-free conjecture for
cyclotomic polynomials under abc conjecture.
