Currently, no feasible (polynomial time) algorithm is known for
recognizing squarefree integers or for computing the squarefree
part of an integer. In fact it may be the case that this problem
is no easier than the general problem of integer factorization.
Computing the radical $\rm\:rad(n)\:$ is equivalent to computing the squarefree part $\rm\:sf(n)\:$ because
$$\rm rad(n)\, =\, sf(n)\, sf(n/rad(n)) $$
This problem is important because one of the main tasks
of computational algebraic number theory reduces to it (in
deterministic polynomial time). Namely the problem of computing
the ring of integers of an algebraic number field depends upon
the square-free decomposition of a polynomial discriminant
when computing an integral basis, e.g.  S.7.3 p.429 or 
This is due to Chistov . See also Problems 7,8, p.9 in ,
which lists 36 open problems in number theoretic complexity.
The primary reason that such problems are simpler in function fields versus number fields
is due to the availability of derivatives. This opens up a powerful
toolbox that is not available in the number field case. For example
once derivatives are available so are Wronskians - which provide powerful
measures of dependence in transcendence theory and diophantine approximation.
A simple yet stunning example is the elementary proof of the polynomial case of Mason's ABC theorem, which yields as a very special case a high-school-level proof of FLT for polynomials, cf.
my MO post and my old sci.math post .
Such observations have motivated searches for "arithmetic analogues of derivations". For example, see Buium's paper by that name in Jnl. Algebra, 198, 1997, 290-99, and see his book Arithmetic differential equations.
 A. L. Chistov. The complexity of constructing the ring of integers
of a global field. Dokl. Akad. Nauk. SSSR, 306:1063--1067, 1989.
English Translation: Soviet Math. Dokl., 39:597--600, 1989. 90g:11170
 Lenstra, H. W., Jr. Algorithms in algebraic number theory.
Bull. Amer. Math. Soc. (N.S.) 26 (1992), no. 2, 211--244. 93g:11131
 Pohst, M.; Zassenhaus, H. Algorithmic algebraic number theory.
Cambridge University Press, Cambridge, 1997.
 Adleman, Leonard M.; McCurley, Kevin S.
Open problems in number-theoretic complexity. II.
Algorithmic number theory (Ithaca, NY, 1994), 291--322,
Lecture Notes in Comput. Sci., 877, Springer, Berlin, 1994. 95m:11142
 Dubuque, Bill. $ $ sci.math.research post, 1996/07/17
poly FLT, abc theorem, Wronskian formalism [was: Entire solutions of f^2+g^2=1]