Find all the Homomorphisms

Question

I want to know how to find all the homomorphisms between $\mathbb{Q}[\sqrt{-3},\sqrt{7}]$ and $\mathbb{C}$ ? Thank you.

Answer 1

Here is a roadmap. If $f: \mathbb{Q}[\sqrt{-3},\sqrt{7}] \to \mathbb{C}$ is a ring homomorphism, then:

• $f(q)=q$ for every $q \in \mathbb{Q}$

• $f(\sqrt{-3})$ is a root of $x^2+3$

• $f(\sqrt{7})$ is a root of $x^2-7$

• $f$ is determined by the images of $\sqrt{-3}$ and $\sqrt{7}$

Indeed, let $\alpha = \sqrt{-3} = 3i$ and $\beta = \sqrt{7}$. Then

$$\mathbb{Q}[\sqrt{-3},\sqrt{7}] = \{ a + b \alpha + c \beta + d \alpha\beta : a,b,c,d \in \mathbb{Q} \}$$ and so $$ f(a + b \alpha + c \beta + d \alpha\beta) =a + b f(\alpha) + c f(\beta) + d f(\alpha)f(\beta) $$ because $f$ is a ring homomorphism that fixes $\mathbb{Q}$. So the question reduces to

How many possible images exist for $\alpha$ and $\beta$ in $\mathbb{C}$ ?