How can I compute the radical of an ideal? I suppose that there no exist an algorithm for compute it. But in the case of polynomials rings? there exist an algorithm? I need to compute the radical of the ideal generated by $ (x^2y , xy^3) $ in the ring of polynomials $ k[x,y]$ where $k$ is any field.
There are indeed algorithms for the polynomial rings, based on the notion of Groebner bases —you can learn about this from David Cox et al.'s book Varieties, Ideals and Alrgorithms.
Those algorithms are actually implemented in several computar algebra systems. I usually use Macaulay2, which google can find for you.
Your ideal $I=(x^2y,xy^3)$ is what's called a monomial ideal, because it is generated by monomials; it has the nice property that a polynomial belongs to $I$ if and only if each of its monomials belongs to $I$, and you can probably prove this easily. It is not hard to see that its radical is also generated by monomials, and then you can probably see which monomials you need in this specific case. I'll let you try your hand at this before giving you the answer...