Exercise of Rotmann Abstract Algebra:
If $E/k$ and $E′/k$ are splitting fields of $f(x) \in k[x]$ and there is a radical extension $K_t/k$ with $E ⊆ K_t$, prove that there is a radical extension $K_{r′} /k$ with $E′ ⊆ K_{r′}$
EDIT:
I am allowed to use:
Context: $B = k(u_1, \dots , u_t)$ finite extension of $k$, with $u_i$ algebraic over $k$. $p_i = \operatorname{irr}(u_i, k) \in k[x]$ the minimal polynomial of $u_i$ over $k$. $f = p_1 \dots pt$. $E$ splitting field of $f$ then:
Lemma 1:
$$E = k(\sigma(u_1), \dots, \sigma(u_t),\sigma \in \operatorname{Gal}(E/k))$$
Lemma 2:
$$u_1^{m_1} \in k, u_2^{m_2} \in k(u_1), \dots , u_t^{m_t} \in k(u_1, \dots, u_{t-1}) \\ \text{ then } E/K \text{ is a radical extension}$$