Field monomorphism from $\mathbb{Q}(\sqrt{5})$ to $\mathbb{C}$.

Question

List (with proof) all field monomorphism from $\mathbb{Q}(\sqrt{5})$ to $\mathbb{C}$.

So I can see that the field monomorphism from $\mathbb{Q}(\sqrt{5})$ to $\mathbb{C}$ are

1. $p+q\sqrt{5}$
2. $p-q\sqrt{5}$

But how do I formally proof this?

Answer 1

Such monomorphism is equal to the identity on $\mathbb Q$. You can verify that the two maps that you have described are indeed field monomorphisms.

Finally there are no others as a field monomorphism sends a root of $X^2-5$ to a root of the same polynomial. And a field monomorphism $\sigma$ of $\mathbb Q(\sqrt{5})$ is totally defined by $\sigma(1)$ and $\sigma(\sqrt{5})$.