I need to determine in which of the cases (a)-(b) the field $F$ is a splitting field of some polynomial $f \in \mathbb{Q}[x]$ over $\mathbb{Q}$?
$$(a)\qquad F = \mathbb{Q}(\sqrt{2}+\sqrt{3}) \\ (b) \qquad \qquad \qquad \quad F = \mathbb{C}$$
I said that the field in $(a)$ is a splitting field, and exhibited the polynomial $f = x^{4}-5x^{2}+6 \in \mathbb{Q}[x]$ to make my point.
For part (b), however, of course, every polynomial $\mathbb{Q}[x]$ splits over $\mathbb{C}$, but I said it was not a splitting field because any polynomial $f \in \mathbb{Q}[x]\setminus \{0\}$ splits completely in the set of algebraic numbers $\mathbb{A}$, which is a proper subset of $\mathbb{C}$ (since, for example, $\pi \in \mathbb{C}$, but $\pi \notin \mathbb{A})$, and by definition, a splitting field is the "smallest" (in terms of inclusion) field over which a polynomial splits, is it not?
However, I am not sure if I answered this question correctly, and was wondering if someone out there might not mind serving as an extra set of eyes to let me know whether I did so, and if not, how I can fix this so that it is answered correctly.
Thank you for your time and patience! :)
Everything looks good.
Once one has shown$^\dagger$ that $\mathbb{Q}(\sqrt{3} + \sqrt{2}) = \mathbb{Q}(\sqrt{3}, \sqrt{2})$, it becomes more obvious that $f(x) = x^4 -5x^2 + 6 = (x^2-2)(x^2-3)$ splits in this field. You might want to discuss why this is the smallest field inside which $f$ splits, but this isn't difficult.
Next, as you've argued, $\mathbb{C}$ cannot be a splitting field for any polynomial: we'll always have $\mathbb{Q} \Big( \{ \alpha_k \}_{k=1}^n \Big) \subsetneq \mathbb{C}$ where the $\alpha_k$'s are the roots of the polynomial, so $\mathbb{C}$ cannot be the smallest field inside which that polynomial splits.
$\dagger$ See arguments here or here for further discussion.