How many field homomorphisms are from $\mathbb{Q}[2^{1/3},2^{1/5}]$ to $\mathbb{C}$.
My idea: Let $\phi:\mathbb{Q}[2^{1/3},2^{1/5}] \rightarrow \mathbb{C}$ be a field homomorphism which preserve the order of $2^{1/3}$ and $2^{1/5}$. i.e $\phi(2^{1/3})^3=2$, and $\phi(2^{1/5})^5=2$.
We have three choices for $2^{1/3}$, and five choices for $2^{1/5}$. Thus there are total $15$ possible choices for $\phi:\mathbb{Q}[2^{1/3},2^{1/5}] \rightarrow \mathbb{C}$.