I am trying to compute the radical of the following ideal: $$ (xy^3+x^3y-x^2+y) $$ in $\mathbb{C}[x,y]$.
My hunch is that this ideal is radical, since it really does not seem amenable to factoring (I have tried...hard), for example, it definitely is not linearly reducible. However, I don't see why this ideal should necessarily kill a variable in the ring showing isomorphism to $\mathbb{C}[x]$ or $\mathbb{C}[y]$ (which would show it is a prime ideal and therefore radical).
Please note, the above is a homework problem so I am only looking for a hint, not a solution. Any reference on methods of tackling problems like the above is welcome. I also do not know the Groebner basis method.