Note that you are not asked to prove that there is a chain of simple irreducible radical extensions $\Bbb Q \subset E_1 \subset E_2 \subset \ldots \subset \Bbb Q(\zeta_n)$, but that there is a chain $\Bbb Q \subset \ldots \subset L$ such that $\Bbb Q(\zeta_n) \subset L$.
If a normal extension $K \subset L$ has Galois Group isomorphic to $\Bbb Z/p \Bbb Z$, then $K(\zeta_p) \subset L(\zeta_p)$ is a normal simple irreducible radical extension :
$K \subset K(\zeta_p)$ is a normal extension, thus $Gal_{K(\zeta_p)}(L(\zeta_p))$ and $Gal_L(L(\zeta_p))$ are normal subgroups of $Gal_K(L(\zeta_p))$ and there is a well-defined restriction map $Gal_K(L(\zeta_p)) \to Gal_K(L) \times Gal_K(K(\zeta_p))$.
This map is injective because $L$ and $\zeta_p$ generate $L(\zeta_p)$. Furthermore, the composition of this map with either projection has to be surjective.
Since $|Gal_K(L)|=p $ and $|Gal_K(K(\zeta_p))|$ are coprime, the only subgroups of their product are products of subgroups, so the map has to be surjective. Thus it is an isomorphism.
let $x \in L \setminus K$. Then $K(x) = L$. Let $\sigma$ be a generator for $Gal_K(L) = Gal_{K(\zeta_p)}(L(\zeta_p))$, and look at $y = \sum_{k=0}^{p-1} \zeta_p^{-k} \sigma^k(x) \in L(\zeta_p)$. Then, $\sigma(y) = \sum_{k=0}^{p-1} \zeta_p^{-k} \sigma^{k+1}(x) = \zeta_p y$.
Thus $\sigma(y^p) = \sigma(y)^p = \zeta_p^p y^p = y^p$. Since $y^p$ is fixed by $\sigma$, $y^p \in K(\zeta_p)$.
Since $X^p - y^p = \prod(X - \zeta_p^k y) = \prod(X - \sigma^k(y))$, this polynomial must be irreducible over $K(\zeta_p)$ (there is exactly one orbit of the action $Gal_{K(\zeta_p)}(L(\zeta_p))$ on its roots)
Finally, $K(\zeta_p,y) = L(\zeta_p)$ because there is an obvious inclusion and their degree over $K(\zeta_p)$ are the same.
Next, if $K \subset L$ is a normal simple irreducible radical extension of degree $p$,
and $K \subset K'$ is a normal extension, then $K' \subset K'L$ is still a normal simple irreducible radical extension, of degree $1$ or $p$. The polynomial $X^p - y^p$ can't suddenly become reducible unless all the $y$ are in $K'$, because $\Bbb Z/p\Bbb Z$ has no nontrivial subgroup.
With this, you can show that if $K \subset L$ and $L \subset M$ are solvable by simple irreducible radical extensions, then so is $K \subset R$ (just add all the roots one after the other).
Finally, with an induction argument, you can finally show that since $K \subset K(\zeta_p)$ is abelian of degree dividing $p-1$, it is solvable, and since $K(\zeta_p) \subset L(\zeta_p)$ is solvable, $K \subset L(\zeta_p)$ is solvable, which implies that $K \subset L$ is too. And by decomposing any abelian extension into cyclic prime extensions, you obtain that any abelian extension is solvable.
Now we apply this to $\Bbb Q \subset \Bbb Q(\zeta_{47})$. Since its Galois group is isomorphic to $\Bbb Z / 2 \Bbb Z \times \Bbb Z / 23 \Bbb Z$, we need to add $\zeta_2$ (which is already there, it's $-1$) and $\zeta_{23}$. For this one, we need $\zeta_{11}$, which needs $\zeta_5$, which needs $\zeta_4$.
The resulting chain (showing the degrees of the extensions) is :
$\Bbb Q \subset^2 \Bbb Q(\sqrt{-1}) \subset^2 \Bbb Q(\sqrt{-1},\sqrt 5) \subset^2 \Bbb Q(\zeta_{20}) \subset^2 \Bbb Q(\zeta_{20}, \sqrt{-11}) \subset^5 \Bbb Q(\zeta_{220}) \subset^2 \Bbb Q(\zeta_{220}, \sqrt{-23}) \\ \subset^{11} \Bbb Q(\zeta_{5060}) \subset^2 \Bbb Q(\zeta_{5060}, \sqrt{-47}) \subset^{23} \Bbb Q(\zeta_{237820}) \supset \Bbb Q(\zeta_{47})$.
Theoretically, you can compute explicitly at each step what are the things you are taking $n$th roots of and express everyone in terms of radicals, though it gets messy really fast.
(I only explicited the quadratic subfields of the $\mathbb Q(\zeta_p)$)
Best Answer
You're right that reducible polynomials need not have roots. For example, take $(x^2 + 1)^2 \in \mathbb R[x]$. However, we may always find a root in some extension of $K$, say the algebraic closure. Indeed, let $\alpha$ be a root of $f$ in an extension of $K$. We would now like to compute $[K(\alpha):k]$.
We have that $[K : k] = m$ by assumption and that $[k(\alpha) : k] = n$ by irreducibility of $f$ over $k$. Then $[K(\alpha) : K] \leq n$ as $\alpha$ is a root of $f$ so its minimal polynomial over $K$ divides $f$ and therefore has lesser degree. Thus, $[K(\alpha) : k] \leq mn$ as $[K : k] = m$ by assumption. Note now that if we prove that this is actually an equality, then we will have shows that $[K(\alpha) : K] = n$. This implies that the minimal polynomial of $\alpha$ over $K$ is degree $n$. As $\alpha$ is a root of $f$, the minimal polynomial of $\alpha$ must divide $f$. As they have the same degree, they must be equal (up to a multiplicative constant), so as the minimal polynomial is irreducible, $f$ must be irreducible.
So it suffices to prove this equality. We can do this by proving the reserve inequality $[K(\alpha) : k] \geq mn$. In fact, we show that $mn \mid [K(\alpha) : k]$. This is where the relative primeness comes into play. As $m$ and $n$ are relatively prime, to prove that $mn \mid [K(\alpha) : k]$ it suffices to prove that $m \mid [K(\alpha) : k]$ and $n \mid [K(\alpha) : k]$. For the first of these, observe that $m = [K : k] \mid [K(\alpha) : k]$. For the second, we have $[k(\alpha) : k] = n$ by irreducibility of $f$ over $k$. Furthermore, $[k(\alpha) : k] \mid [K(\alpha) : k]$ so we are done.