How do i find the minimal polynomial over the field k?

Question

I have a field $k$ and $x$ is a variable, where $y=\frac{x^3}{x+1}$ how do I find the minimal polynomial of x over $K(y)$

I'm quite confused, I'm trying to use the theorem on primitive elements, so far I have:

$x^3-(x+1)y=x^3-\frac{(x+1)y}{x+1}=0 $

where do I go from here, to show that x is a minimal polynomial over $k(y)$?

any help would be appreciated.

Answer 1

$x^3-yx+y\in K[y][x]$, a polynomial ring over the P.I.D. $K[y]$. Eisenstein's criterion shows this polynomial is irreducible, hence it is the minimal polynomial.

Answer 2

Another way: apply the Rational Root Test to conclude that if $\,x^3 - x y + y\,$ is reducible over $\,k(y)\,$ then it has a root $\in k[y]\,$ which is a factor of the prime $y,\,$ contra a simple degree argument.