Suppose $K$:= $\mathbb{Q}(\sqrt[3]{5}, \sqrt[4]{2})$

Explicitly find the number of elements in Hom$(K, \mathbb{R})$ and Hom$(K, \mathbb{C})$

Normally I would find the primitive element, $\alpha$, such that $K$ = $\mathbb{Q}(\alpha)$. Then just look at how many real and non real roots are there in the minimum polymial of $\alpha$.

But here I'm looking at a potential degree 12 minimum polynomial, and hence hard to compute.

Thus I feel that there are some easier method that I'm not seeing here.

Any help or insight is deeply appreciated.


Hint: If $\sqrt[3]{5} \mapsto \alpha$, the $\alpha$ must satisfy the polynomial $x^3-5$ over $\mathbb{R}$ (and $\mathbb{C}$). Similarly for $\sqrt[4]{2}$. Try to see how many solution you can have over $\mathbb{R}$ (and $\mathbb{C}$).


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