Degree of extension $\Bbb Q(\sqrt[4]{5},\sqrt[6]{7})$.
The polynomial $x^4-5$ can be factored over $\mathbb{R}$, which contains $\mathbb{Q}(\sqrt[6]{7})$, as $$ x^4-5=(x-\sqrt[4]{5})(x+\sqrt[4]{5})(x^2+\sqrt{5}) $$ Thus a factorization over $\mathbb{Q}(\sqrt[6]{7})$ can only be the one above or $(x^2-\sqrt{5})(x^2+\sqrt{5})$. So you just need to show that $\sqrt{5}\notin\mathbb{Q}(\sqrt[6]{7})$.
Suppose $\mathbb{Q}(\sqrt{5})\subseteq\mathbb{Q}(\sqrt[6]{7})$. Then the degree of $\sqrt[6]{7}$ over $\mathbb{Q}(\sqrt{5})$ is $3$ by the dimension formula. The factorization of $x^6-7$ over $\mathbb{R}$ is $$ (x^3-\sqrt{7})(x^3+\sqrt{7})= (x-\sqrt[6]{7})(x^2+\sqrt[6]{7}\,x+\sqrt[3]{7}) (x+\sqrt[6]{7})(x^2-\sqrt[6]{7}\,x+\sqrt[3]{7}) $$ Since $\sqrt{7}\notin\mathbb{Q}(\sqrt{5})$, you can only get degree three factors as $$ (x-\sqrt[6]{7})(x^2-\sqrt[6]{7}\,x+\sqrt[3]{7}) $$ or $$ (x+\sqrt[6]{7})(x^2+\sqrt[6]{7}\,x+\sqrt[3]{7}) $$ and in both cases you'd conclude that $\sqrt[6]{7}\in\mathbb{Q}(\sqrt{5})$, which is impossible.
Suppose $x^4-5$ can be factored as $f(x)g(x)$ over some extension $K$ of $\mathbb{Q}$, $K\subseteq\mathbb{R}$; suppose also that $f(x)$ and $g(x)$ are non constant. Since $x^4-5$ is monic, also $f$ and $g$ can be assumed monic. Continuing like this, we can assume that $x^4-5$ is factored into monic factors, irreducible over $K[x]$.
Let $h(x)\in K[x]$ be one of these factors; its factorization in $\mathbb{R}[x]$ must consist of polynomials in the set $\{x-\sqrt[4]{5},x+\sqrt[4]{5},x+\sqrt{5}\}$, which are the irreducible factors of $x^4-5$ in $\mathbb{R}[x]$, because of uniqueness of factorization in $F[x]$ (for $F$ any field).
Now it's just a matter of checking the various possibilities. A factorization of $x^4-5$ can only be with degrees
- $1$, $1$ and $2$
- $2$ and $2$
- $1$ and $3$
If a degree $1$ factor appears, then $\sqrt[4]{5}\in K$; if a degree $2$ factor appears, then $\sqrt{5}\in K$. In both cases, $\sqrt{5}\in K$.
The same argument applies for the second part of the proof.
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}) $.