This is not an answer to my question but it is just a summary of the relevant results  which have been mentioned so far about **positive solvability**.

So here is a useful result:

**Theorem** Let $K/F$ be a finite Galois extension and let $M/F$ be a radical extension.
Assume that you have an embedding $\iota:K\hookrightarrow M$. Then if $[K:F]$ is odd there exists a root of unity of odd order inside $M$.    

A proof of this result may be found for example in Brian Conrad's note: [radical tower and roots of unity][1].

We thus have the following corollary

**Corollary** Let $f(x)\in\mathbf{Q}[x]$ be an irreducible polynomial of odd degree
which has all its roots in $\mathbf{R}$. Let $K$ be the (abstract) splitting field of $f(x)$. Then
$K$ cannot be embedded in a radical extension contained in $\mathbf{R}$. In particular,
$f(x)$ is not positive solvable.

 







[1]:http://math.stanford.edu/~conrad/210BPage/handouts/radreal.pdf