# confusion about the difference between the $\Bbb R$-minimal polynomial of $\alpha$ and the $\Bbb Q$-minimal polynomial

I am confused by the terminology in an old assignment I was looking at, the question is as follows:

Find the $$\Bbb R$$-minimal polynomial of $$\alpha$$ and then ﬁnd the $$\Bbb Q$$-minimal polynomial of $$\alpha$$.

Where $$\alpha:=\sqrt{5}+i$$

I know how to find minimum polynomial given roots, in this case, we have :

let

$$x=\sqrt{5}+i$$

then

$$x^2=5+2i\sqrt{5}-1=4+2i\sqrt{5}$$

so

$$x^2-4=2i\sqrt{5}$$

therefore

$$(x^2-4)^2=-20$$

giving our min. poly as

$$x^4-8x^2+36$$

The part that confuses me is the difference between the $$\Bbb R$$-minimal polynomial of $$\alpha$$ and the $$\Bbb Q$$-minimal polynomial of $$\alpha$$ is and furthermore what is the difference in calculation between the two?

$$x^2-2x\sqrt5+6$$ is the minimal polynomial of $$\alpha$$ over $$\mathbb R$$. Note that a minimal polynomial of $$\alpha$$ over some field $$\mathbb F$$ is an element of $$\mathbb F[x]$$. So over $$\mathbb R$$, we can have an irrational number as a coefficient of the minimal polynomial, whereas it is not allowed over $$\mathbb Q$$.
• Oh so we just do $(x-\sqrt{5})^2=i^2$ ? – excalibirr Jun 14 at 17:16