I have the following problem:
Let $k=\Bbb{F}_2(X)$ and $E=\Bbb{F}_2(\sqrt{X})$. Then I know that the minimal polynomial of $\sqrt{X}$ over $k$ is $t^2-X$. But now in the lecture our prof. says that this polynomial splits into $(t-\sqrt{X})^2$ over $E$ and thus only has one root.
But I somehow don't see why it splits into $(t-\sqrt{X})^2$ and not into $(t-\sqrt{X})(t+\sqrt{X})$. As I know it we define $$\Bbb{F}_2(\sqrt{X})=\{a+b\sqrt{X}:a,b\in \Bbb{F}_2\}$$ Then I would say that $-\sqrt{X}$ is equivalent to $\sqrt{X}$ since $[-1]=[1]$ in $\Bbb{F}_2$ but I'm not sure if this is enought to conclude that $(t-\sqrt{X})(t+\sqrt{X})=(t-\sqrt{X})^2$.
Could someone maybe help me.