Can two splitting fields over the same field of a polynomial be different (in the set sense and not up to isomorphism)?

Question

Consider the polynômial $x^2-3$ over $\mathbb{Q}$. A splitting field would be $\mathbb{Q}(\sqrt{3})$. I also know, via some theorem that if I have another splitting field $S$ over $\mathbb{Q}$), it should be isomorphic to $\mathbb{Q}(\sqrt{3})$ and the isomorphisme $\phi$ is identity on the elements of $\mathbb{Q}$. I understand it should be at least isomorphic, but can S be different of $\mathbb{Q(\sqrt{3})}$ (in the set equality sense, not the isomorphic sense)? I thought of this because it is not specified that the isomorphism is the identity on $S \setminus \mathbb{Q}$.

Answer 1

$\mathbb{Q}(\sqrt{3}) \subseteq \mathbb R$ and $\mathbb{Q}[x]/(x^2-3)$ are different sets.