Problem
Assume $F$ contains the $k$th roots of unity, and let $R=F(\alpha)$, where $\alpha$ is a root of $x^k-a$ for some $a\in F$. Prove that there exist intermediate fields $$F=K_0\subset K_1\subset \cdots\subset K_t=R$$
where $K_{i+1}=K_i(\beta_i)$, where $\beta_i$ is a root of $x^{p(i)}-b_i$ for some $b_i\in K_i$ and $p(i)$ is a prime.
My thoughts
The problem itself really doesn't offer any real hints as to the idea it is exactly pointing at. But I am assuming that it really is saying we can find intermediate "prime" fields (where each prime divides $k$) that essentially build this root field.
An example of what I think the problem means:
Let $F=<\mathbb{Q}\bigcup \{e^{\frac{2\pi in}{6}}\}>$, $\alpha$ be the root of $x^6-3$, $K_1=K_0(3^{1/2})$ and $K_2=K_1(3^{1/3})$ then
$F=K_0\subset K_1\subset K_2=R$
this is true mainly because $\frac{1}{3}3^{\frac{2}{3}}3^{\frac{1}{2}}=\frac{1}{3}3^{\frac{4+3}{6}}=\alpha\in K_2$
and
Let $F=<\mathbb{Q}\bigcup \{e^{\frac{2\pi in}{8}}\}$>, and $\alpha$ be a root of $x^8-2$ then let $K_1=K_0(2^{\frac{1}{2}})$, $K_2=K_1(2^{\frac{1}{4}})$, and $K_3=K_2(2^{\frac{1}{8}})$ it follows that
$F=K_0\subset K_1\subset K_2\subset K_3=R$
My question is: Is the idea they are trying to get across in the question the fact:
Assuming $a\in K_0$, it follows that if $k=p_1^{e_1}p_2^{e_2}...p_n^{e_n}$, then $a^{\frac{1}{k}}=(...((a)^{\frac{1}{p_1^{e_1}}})^{\frac{1}{p_2^{e_2}}}...)^{\frac{1}{p_n^{e_n}}}$. So we can then create fields $K_0\subset ...\subset K_t$ where for the first $K_0\subset ...\subset K_{e_1}$ have the property that $K_{i+1}=K_i(a^{\frac{1}{p_1^{i+1}}})$ and the next couple fields $K_{e_1}\subset ...\subset K_{e_1+e_2}$ have the property that $K_{e_1+i+1}=K_{e_1+i}(a^{\frac{1}{p_1^{e_1}p_2^{i+1}}})$ and so on until we get to $K_t$ where $t=\sum\limits_{i=1}^{n}e_i$
but without referring to the elements $a^{\frac{1}{r}}$ (I have noticed a trend of the book saying: "let $\alpha$ be a root of $x^2-2$," rather than saying, "let $\alpha=\sqrt{2}$.")? If not what am I supposed to be seeing in this problem?
Notation
$<A>$="Smallest field containing A"