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

Question

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?

Answer 1

$x^2-2x\sqrt5+6$ 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 $\mathbb Q$.