$k[X]$ is integral over $k[X^{2}]$

The easiest way is to note that $\,x\,$ is integral over $\,k[x^2],\,$ being a root of $\,y^2 - x^2.\,$ For a more direct proof, bisect $\,y\in k[x]\,$ into even+odd part $\, y = a + b\:\! x,\ a,b\in k[x^2],\,$ so $\, (y-a)^2 = b^2 x^2\,$ shows $\,y\,$ is a root of a monic quadratic over $\,k]x^2],\,$ so $\,y\,$ is integral over $\,k]x^2]$.