Compositum of solvable extensions

Question

i have a question on solvable extensions.

I am reading a paper where the authors work with the field $\mathbb{Q}^{solve}$, that is, the maximal solvable extension of $\mathbb{Q}$ in $\overline{\mathbb{Q}}$. I am not sure on what is the definition of this field, but my guess is that is the compositum of all solvable extensions of $\mathbb{Q}$. However, i am not so sure if the compositum of two solvable extensions is still solvable.

Is this true? If not, do you have a counterexample?

Thanks!

Answer 1

Hint : Note that solvable extensions form a distinguished class of field extensions.

Now look at the set :

$ \mathbb{S}$ = { x $\in \mathbb{C} $ : x is algebraic and the Galois Group of $ {min}_{\mathbb{Q}}(x)$ is solvable }