$X$ an integral element over $K[X^d]$

Question

I'm currently redoing some homework for my algebra test and I came across a task that wants me to find a noether normalization of $k[X]$, where $k$ is a field. The proof I know uses the argument, that $X$ is an integral element over $k[X^d]$ for $d$ element of IN. However I don't remember why exactly that is the case. I'd be very thankful for some advices regarding this problem, thank you in advance for answers!

Answer 1

It is a solution of the monic polynomial equation $$y^d-X^d=0$$ where the indeterminate is $y$ and $X^d$ is the element of the ring.