I already showed out that the four numbers: $\pm \sqrt 2 \pm \sqrt 3$ are the root of the polynomial $p(x) = x^4 -10x^2 +1$.
In addition, I showed that $p(x)$ is irreducible in $\mathbb Q [X]$
Are the two claims sufficient to conclude that the degree of those four numbers is $4$? If not, how do I show that?
Thanks
Quite generally, if $k$ is a field and $f(X)\in k[X]$ is irreducible in that ring and has degree $n$, then any root $\rho$ of $f$ will be of degree $n$ over $k$, if by that you mean that $\bigl[k(\rho):k\bigr]=n$. You claim that you’ve proved both of the required properties, so yes, your numbers are of degree four over $\Bbb Q$.