[Math] Minimal polynomial of $\sqrt2+1$ in $\mathbb{Q}[\sqrt{2}+\sqrt{3}]$

Question

I'm trying to find the minimal polynomial of $\sqrt2+1$ over $\mathbb{Q}[\sqrt{2}+\sqrt{3}]$. The minimal polynomial of $\sqrt2+1$ over $\mathbb{Q}$ is

$$ (X-1)^2-2.$$

So I look at $\alpha = \sqrt2 + \sqrt3$

$$ \alpha^0 = 1$$
$$ \alpha^1 = \sqrt2 + \sqrt3.$$

I cannot find $a,b$ such that $a\alpha + b\alpha = \sqrt2+1$. So the degree 2 is mimimal. Is that correct?

Answer 1

Letting $t = \sqrt{2} + \sqrt{3}$, $\sqrt{2} + 1 = 1 + \frac{1}{2}(t^3 - 9t)$ so the minimal polynomial is $x - \sqrt{2} - 1$.