factors that are naturally associated to every integer: the square-free part, the above factor , and the largest square-free factor. Each is a factor of the next one. All are easily deduced from the prime factorization or the square-free factorization: if are the prime factorization and the square-free factorization of , where are distinct prime numbers, then the square-free part is The square-free factor such the quotient is a square is and the largest square-free factor is For example, if one has The square-free part is , the square-free factor such that the quotient is a square is , and the largest square-free factor is . No algorithm is known for computing any of these square-free factors which is faster than computing the complete prime factorization. In particular, there is no known polynomial-time algorithm for computing the square-free part of an integer, or even for determining whether an integer is square-free. In contrast, polynomial-time algorithms are known for primality testing. This is a major difference between the arithmetic of the integers, and the arithmetic of the univariate polynomials, as polynomial-time algorithms are known for square-free factorization of polynomials (in short, the largest square-free factor of a polynomial is its quotient by the greatest common divisor of the polynomial and its formal derivative). Equivalent characterizations A positive integer is square-free if and only if in the prime factorization of , no prime factor occurs with an exponent larger than one. Another way of stating the same is that for every prime factor of , the prime does not evenly divide . Also is square-free if and only if in every factorization , the factors and are coprime. An immediate result of this definition is that all prime numbers are square-free. A positive integer is square-free if and only if all abelian groups of order are isomorphic, which is the case if and only if any such group is cyclic. This follows from the classification of finitely generated abelian groups. A integer is square-free if and only if the factor ring (see modular arithmetic) is a product of fields. This follows from the Chinese remainder theorem and the fact that a ring of the form is a field if and only if is prime. For every positive integer , the set of all positive divisors of becomes a partially ordered set if we use divisibility as the order relation. This partially ordered