The below is from chapter 14.7 of Dummit & Footes "Abstract Algebra" (paraphrased).
Theorem: Let $\alpha \in K$ for $K$ a root extension. Then $\alpha$ is contained in a root extension which is Galois over $F$ (the base field) and where each extension $K_{i+1}/K_i$ is cyclic.
In the proof, there is a claim that:
It is easy to see that the composite (field) of two root extensions is a root extension (take $K'$ another root extension, with subfields $K'_i$, first take the composite of $K'_1$ with $K_0,K_1,\ldots,K_s$, and then take the composite of these fields with $K'_2$ etc., so that each individual field in this process is a simple radical extension).
My understanding is that this is the process: We start by creating the composites $K_0K'_1,\ldots,K_0K'_1$, then we take the composite of these with $K'_2$ and get $K_0K'_1K'_2,\ldots,K_sK'_1K'_2$ and so on up until we get $K_0K'_1K'_2 \cdots K'_{s'},\ldots,K_s K'_1 K'_2 \cdots K'_{s'} = KK'$ (where the last equality follows from the fact that $K'_i \subseteq K'_j$ for $i < j$).
First of all, I don't understand why we successively do this process as described, since I believe e.g. $K_0K'_1K'_2 = K_0K'_2$ since $K'_1 \subset K'_2$. Secondly, I don't understand what the chain terminating in $KK'$ is. Like, if we start with $F = K_0 \subset K_0K'_1 = K'_1 \subset K_1K'_1 \subset \cdots \subset K_sK'_1 = KK'_1$, then how do I continue the chain here?
Have I misunderstood Dummit & Footes description of this process?
