I'm studying Galois theory by Morandi's book 'Field and Galois Theory'. I've found an exercise that investigate when a rational polynomial is solvable by 'real radicals'. It is possible to show that a rational polynomial with only real roots is solvable by real radicals if and only if $[N:Q]=2^n$, where $N$ is the splitting field of $f$ over $Q$.
I would like to know if there is any way to prove if a rational polynomial has only real roots or not. Thank you.
