Prove or disprove: $\Bbb Q (\sqrt{3})$ is isomorphic to $\Bbb Q (-\sqrt{3})$.

Question

I think that $\Bbb Q (\sqrt{3})$ is isomorphic to $\Bbb Q (-\sqrt{3})$ but I do not know how to show this.

Answer 1

Hint: $$a+b\sqrt{3}=a+(-b)(-\sqrt{3})$$

Answer 2

If you can prove they're both the same field, then the identity function from one of them to the other is an isomorphism.

You can prove they're the same if you can prove that every field that includes $\mathbb Q$ and contains $\sqrt 3$ also contains $-\sqrt 3$ and every field that includes $\mathbb Q$ and contains $-\sqrt 3$ also contains $\sqrt 3$.

You should also be able to show that there is an isomorphism that maps $\pm\sqrt 3$ to $\mp\sqrt 3$.