I'm reading page $59$ of Reid's "Undergraduate commutative algebra" book.
In example (ii) it says, $k[x^{2}] \subset k[x]$ is an integral extension.
How do we know this? I mean, in order to show this we must take a polynomial $f(x) \in k[x]$ and show there is a monic polynomial $g(x) \in k[x^{2}]$ such that $g(f(x))=0$, right? Why can we do this?