Solubility of a Galois Group

going over some past papers with no answers and would like a bit of help if possible..

I've shown that for p a prime number then $x^p-1 \in K[x]$ is abelian where K is a subfield of $\mathbb{C}$. I've now been asked to show that the Galois group of $x^p-a$ over K is soluble with $a \neq 0 \in K$. I know any abelian group A is soluble, since ${1} \triangleleft A$ is a subnormal series with its only factor A being abelian, so $x^p-1 \in K[x]$ is certainly soluble. Not sure where to go from here and the question is worth quite a lot so guessing there is quite a bit more to do, any help appreciated.

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2022-07-25 20:43:41
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