TITLE: Degree of extension $\Bbb Q(\sqrt[4]{5},\sqrt[6]{7})$.
QUESTION [6 upvotes]: I tried to consider the tower of extension $\Bbb Q\subset \Bbb Q(\sqrt[6]{7})\subset\Bbb Q(\sqrt[4]{5},\sqrt[6]{7})$.
The minimal polynomial of $\Bbb Q(\sqrt[6]{7})$ over $\Bbb Q$ is $x^6-7$ by Eisenstein. But although it is easy to see that it has no root in $\Bbb Q(\sqrt[6]{7})$, how can I formally conclude that $x^4-5$ is irreducible over $\Bbb Q(\sqrt[6]{7})$ and thus we can see the basis of $\Bbb Q(\sqrt[6]{7}, \sqrt[4]{5})$？
I know that for if we have the degree of $\Bbb Q(\sqrt[6]{7})$ and $\Bbb Q(\sqrt[4]{5})$ are coprime, then it can be much simpler. But how to deal with that in this case where they are not coprime. Any help would be appreciate. Thanks so much！

REPLY [2 votes]: In general, we have the following: Let $ K/\mathbf Q $ and $ L/\mathbf Q $ be two number fields such that there exists a rational prime $ p $ which is totally ramified in $ K/\mathbf Q $ and unramified in $ L/\mathbf Q $. Then, the extensions $ K/\mathbf Q $ and $ L/\mathbf Q $ are linearly disjoint, that is, $ [KL : \mathbf Q] = [K : \mathbf Q][L : \mathbf Q] $.
Proof. Let $ \mathfrak q $ be a prime of $ LK $ lying over the rational prime $ p $. Then, $ e_{\mathfrak q | p} \geq [K : \mathbf Q] $, since ramification indices are multiplicative across towers and $ p $ is totally ramified in $ K/\mathbf Q $.  On the other hand, if we let $ \mathfrak p $ be the prime of $ L $ lying below $ \mathfrak q $; then we have that 
$$ [K : \mathbf Q] \leq e_{\mathfrak q | p} = e_{\mathfrak q | \mathfrak p} e_{\mathfrak p | p} = e_{\mathfrak q | \mathfrak p} \leq [LK : L] $$
However, we obviously have that $ [K : \mathbf Q] \geq [LK : L] $; thus it follows that $ [K : \mathbf Q] = [LK : L] $, and multiplication by $ [L : \mathbf Q] $ on both sides of the equality gives the result.
Now, notice that we have exactly the situation of this claim with the rational prime $ p = 5 $, which is totally ramified in $ \mathbf Q(\sqrt[4]{5}) $ but unramified in $ \mathbf Q(\sqrt[6]{7}) $.