A Bring radical of $a$ is any root of the polynomial $x^5+x+a$. It is known that we can solve the quintic if we're allowed to use the Bring radical.
Now, I was wondering what happens if we generalize: define the $n$th Bring radical of $a$ to be some root of $x^n+x+a$. Can all higher degree polynomials by solved by radicals and Bring radicals? if not, is there a criterion on the Galois group that is equivalent to solvability by radicals and generalized Bring radicals?