Show that a polynomial is solvable (soluble)

Question

So I've recently encounter the problem:

Show that the Galois group of $x^{10}-2$ over Q is solvable

I've been trying tp show that the splitting field of this pol is contained in a radical extension but it didn't turn out right.

Is there anyway to approach this type of problem?

Answer 1

$\Bbb Q \subseteq \Bbb Q(\zeta_{10}) \subseteq \Bbb Q(\zeta_{10}, 2^{1/10})$.

The first extension is cyclotomic hence abelian.

The second extension is cyclic.

Answer 2

The roots of this polynomial equation are expressible as $x=\root 10\of 2$, hence, by definition, as it is expressed as a radical in the coefficients, it is solvable by radicals.