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

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 $$S \setminus \mathbb{Q}$$.

• Take a splitting field $K$, where all elements have the same color as the rational numbers, which (we know) are pale green; then paint red the elements of $K\setminus\mathbb{Q}$. You have a very simple example of “different” splitting fields. – egreg Sep 28 '18 at 13:58

$$\mathbb{Q}(\sqrt{3}) \subseteq \mathbb R$$ and $$\mathbb{Q}[x]/(x^2-3)$$ are different sets.
• Yes, but I was wondering, for subsets $S$ and $S'$ that contain $\mathbb{Q}$ and not just imbed it. – roi_saumon Sep 28 '18 at 14:09