I want to show that $\sqrt[p]{\ell}\notin L:=\mathbb{Q}[\zeta_p,\sqrt[p]{5}]$ where $\ell$ and $p$ are primes and $\zeta_p$ a primitive $p$-th root of unity $(\zeta_p)^p=1$.
I edited my post since what I wrote wasn't satisfying. The route I followed was sterile and the trick is to stay over $K_p:=\mathbb{Q}[\zeta_p]$ since we can then apply the "Kummer" trick, no need for more sophisticated tools. Suppose for a contradiction that $\mathbb{Q}[\zeta_p,\sqrt[p]{5}]=\mathbb{Q}[\zeta_p,\sqrt[p]{\ell}]$. We write $\alpha=\sqrt[p]{5}$, $\beta=\sqrt[p]{\ell}$ and $K=K_p[\alpha]=K_p[\sqrt[p]{\ell}]$ to emphasis the point. Then one can write:
$\beta=c_0+c_1\alpha+c_2\alpha^2+\cdots+c_{p-1}\alpha^{p-1}$ whit $c_i\in K_p$ (1)
Take a generic automorphism $g$ from $\mathrm{Gal}(L/K_p)$ defined by $g(\alpha)=\zeta\alpha$. What I call the Kummer trick is to see that $g(\beta)/\beta$ is a $p$-th root of unity ($\big(g(\beta))^p=g(\beta^p)=g(\ell)=\ell=\beta^p$ since $\ell$ is invariant.
So $g(\beta)=\zeta^k\beta$ for some $k$ (2)
Then you compose $(1)$ by $g$ and use (2):
$g(\beta)=c_0+c_1\zeta g(\alpha)\cdots+g(\alpha^{p-1})=c_0+c_1\zeta^r\alpha\cdots+\zeta^r\alpha^{p-1}$ and you find that $\beta=c_{i_0}\alpha^{i_0}$, which raised to the $p$-th power gives $\ell=q5^r$, a contradiction.
I took this from K. Conrad's Linear independence of characters.
@user26857 Let $K = \mathbf Q(\zeta_p)$ and $L = K(\sqrt [p] {5})$ . The question amounts to show that $ L\neq K(\sqrt [p] {l})$ except in an obvious particular case. In your « Kummer trick », the mix of the additive and multiplicative structures complicates things. Besides, in your final equation $l = q. 5^r$, if $q$ is merely in your field $K_p$ , I can’t see a contradiction . Can you elaborate ?
The ghist of Kummer theory is to use the multiplicative structure alone. Because $K$ contains $\zeta_p$, Kummer says that $ L = K(\sqrt [p] {l})$ iff the two subgroups of $K^*$/ ($K^*)^p$ generated by $5$ and $l$ coincide, iff there exists $1\le i\le p-1$ such that $5.l^i = x^p$ , $x\in K^*$. Since $5$ and $l$ are in the ring of integers $A := \mathbf Z[\zeta_p]$ , $x$ must be also in $A$. Norm down to $\mathbf Z$ to get $5^{p-1}.l^{i(p-1)} = y^p$. Excluding the particular case where $l = 5$ , this equation is impossible because of the exponents and of the factoriality of $\mathbf Z$ .